\section{Introduction}

Let $F:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m_0 \rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m_0$ be a germ of a
holomorphic map with $F(0)=0, F'(0)={\mbox{Id}}.$
Dominique Cerveau ([C]) asked whether $F$ can have a dense
orbit. More precisely, does there exist a small ball
${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,\delta)={\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_\delta=\{z\in {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m, \|z\|<\delta\}$ where $F$ is defined and a sequence
$S=\{p_n\}_{n=0}^\infty \subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,\delta)$,
$F(p_n)=p_{n+1}\; \forall \; n$,  so that
$S \cap {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,\epsilon)$ is dense in some smaller ball
${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,\epsilon), 0<\epsilon<\delta.$
Our main result is that this is not possible. We prove in fact a
more general result. Recall that a domain $\Omega$
in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^{m}$is said
to have a Lipschitz boundary if we can locally write,
after a holomorphic rotation of coordinates,
$\Omega=\{{\mbox{Im}}z_{m}< r(z_1,\dots,z_{m-1},{\mbox{Re}}z_{m})\}$ where
$|r(x)-r(y)|\leq C \|x-y\|,$ the constant $C$ is called a Lipschitz constant
for $r.$

\begin{main theo}
Let $\Omega \subset \subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ be a domain with
a Lipschitz boundary. Suppose that $F:\Omega
\rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ is a holomorphic map and let $V \subset \Omega$
be a nonempty open subset. Then $F$ does not have a dense orbit
in $V$, i.e. there does not exist a sequence
$\{p_n\}_{n\geq 0}\subset \Omega$
for which $F(p_n)=p_{n+1}$ for all $n$ and
$\{p_n\}\cap V$ is dense in $V.$
\end{main theo}

The condition that $\Omega$ has Lipschitz boundary is not necessary,
see Remark 5.4. However, the following theorem shows that
the result is false if we have no boundary conditions.

\begin{theo}
There is a bounded domain $U$ in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ which is biholomorphic
to the unit ball and a holomorphic
map $F:U \rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ with a dense orbit. So there is a dense sequence
$\{p_n\}_{n=0}^\infty \subset U$ with $F(p_n)=p_{n+1}$
for all $n.$
\end{theo}

The proofs depend on two results which are of independent
interest.

\begin{theo} Suppose that $\Omega \subset \subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$
is a domain with Lipschitz boundary. Then if $p\in \partial
\Omega$ there are two not mutually exclusive possibilities:\\
\noindent (i) The point $p$ is in the envelope of holomorphy of $\Omega.$\\
\noindent (ii) The point $p$ is in a local peak set for plurisubharmonic
functions, i.e. there exists a neighborhood $U$ of $p$ and
a continuous function $\rho: U \cap \overline{\Omega}\rightarrow
\RR$ so that $\rho<0$ on $U \cap \Omega$ and $\rho$ is plurisubharmonic
on $U \cap \Omega$. Moreover, $\rho(p)=0.$
\end{theo}

\begin{theo}
Let $M^m$ be an $m-$dimensional
Hausdorff connected complex manifold with countable topology. Then
there exists a biholomorphic map $F: {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m(0,1) \rightarrow U$
from the unit ball onto an open subset $U \subset M$ so that
$M\setminus U$ has zero volume.
\end{theo}

We prove Theorem 1.4 in Section 2. The proof relies heavily on
previous joint work of the first author and E. L. Stout. They
proved, [FS1], that Theorem 1.4 is valid if one replaces the ball
with the polydisc of the same dimension. They also proved
covering theorems by balls, [FS2], in arbitrary complex manifolds.
We would like to thank E. L. Stout for valuable
remarks during the work with Theorem 1.4.

\bigskip

Theorem 1.3 is proved in Section 3. The two main ingredients is
an estimate of the local envelope of holomorphy of a domain
with Lipshitz boundary together with the result by Demailly [D]
that Lipschitz pseudoconvex domains are hyperconvex, $i.e.$
that they allow a bounded plurisubharmonic exhaustion function.
The key result about envelopes of holomorphy is the following:

\begin{theo}
Let $r(z_1,\dots,z_{m-1},{\mbox{Re}}(z_{m}))$ be a real-valued
Lipschitz function on ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^{m-1}\times \RR$ with Lipschitz constant
$C$. Set
$$\Omega:=\{{\mbox{Im}}(z_{m})<r(z_1,\dots,z_{m-1},{\mbox{Re}}(z_{m}))\}.$$
Then the envelope of holomorphy $\tilde{\Omega}$ of $\Omega$
is schlicht and is of the form
$\tilde{\Omega}:=\{{\mbox{Im}}(z_{m})<\tilde{r}
(z_1,\dots,z_{m-1},{\mbox{Re}}(z_{m}))\},$
where $\tilde{r}\geq r$ is a Lipschitz function with the same
Lipshitz constant or $\tilde{r}\equiv \infty.$
\end{theo}

We prove Theorem 1.2 in Section 4. The proof goes by an explicit
construction of an expanding  map defined on a biholomorphic image of
the polydisc, and then using Theorem 1.4 to find a dense open subset
in this domain, biholomorphic to the ball.

\bigskip

Finally we prove the Main Theorem in Section 5 by contradiction.
A key step is to use Theorem
1.3 to show that the iterates $F^n$ are well defined on $V$ with
image contained in $\overline{\Omega}.$ The key point is to show that
$F^n(V)$ is contained in $\tilde{\Omega}$ so that $F^{n+1}$ has a holomorphic
extension.This allows
us to use normal families arguments.

\section{Proof of Theorem 1.4}

To prove Theorem 1.4 it suffices by Theorem 1 of [FS1] to assume that
$M=\Delta^m$. In fact we will prove the Theorem in the
slightly more general case, when $M=U$, a connected open set
in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$. This is anyhow necessary as we change $U$ during the proof.

\begin{lem}
Let $U$ be a connected open set in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$. Suppose
$F_j:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j} \rightarrow U, j=1,2$ are $1-1$ holomorphic maps
with $F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_1}) \cap F_2({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_2}) =\emptyset.$
Let $0<s<1$. Then there exists a $1-1$ holomorphic map
$F_0:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$ so that $F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)
\supset F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{sr_1})\cup F_2({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{sr_2}{32}})$
\end{lem}

\noindent{\bf Proof.} If $a$ is a real number, we shall understand by
$\bf a$ the point
$(a,0,\ldots,0)$ in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$. Also, for a positive number $t$,
let $K_t$ be the union of the closed ball $\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1}$,
the closed ball ${\mathbf 4}+\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_t}$ and the straight
line segment $[\mathbf 0,\mathbf 4]$. The set $K_t$ is thus a certain
compact subset of ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$, which is polynomially convex when $t<3$.
Fix a number $\sigma\in(s,1)$. Let $p\in F_1(b{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_1})$
be the point
$F_1({\mathbf{\sigma r_1}})$
and $q\in F_2(b{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_2})$ the point $F_2(-{\mathbf{\sigma r_2}})$.
Let $\lambda$ be a smooth arc in $U$ that connects $p$ and $q$ and is
otherwise disjoint from $F_1(\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_1}})
\cup F_2(\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_2}})$. Choose a smooth parametrization
$f:[{\bf 1}, {\bf 2}]\rightarrow \lambda$ that takes $\bf 1$ to
the point $p$ and $\bf 2$ to $q$.
The map $G:K_2\rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ is defined by
\begin{equation*}
G(z)=\begin{cases}
F_1(\sigma r_1 z)& \mbox{if $z\in\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1}$}\\
F_2(\frac{\sigma r_2 }{2}(z-{\mathbf{4}}))&
\mbox{if $z\in{\mathbf{4}}+\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_2}$}\\
f(x)& \mbox{if $z={\mathbf x}$ with $x\in[{\bf 1},{\bf 2}].$}\\
\end{cases}
\end{equation*}

With the proper choices of $\lambda$ and $f$, the function $G$ defined in
this way is of class ${\mathcal {C}^\infty}$ on$[{\mathbf {0}},
{\mathbf{4}}]$. It is one-to-one on $K_2$.
The function $G$ can be approximated in the ${\mathcal{C}^1}$ sense on the
set $K_2$ by polynomial maps $P:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m\rightarrow{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ [FS1].
If $P$ is such a map that approximates $G$ well enough, then $P$ will be
one-to-one on $K_2$ and can be chosen to be regular on a neighborhood of
$K_2$. It will satisfy also the conditions

\noindent a) $P(K_2)\subset U$,

\noindent b) $P(K_2)\supset F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{sr_1})\cup F_2({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{sr_2})$, and

\noindent c) $P\left({\mathbf{4}}+{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_\frac{1}{16}\right)\supset
F_2\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{sr_2}{32}}\right)$.

Let $V\subset{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ be a neighborhood of $K_2$ on which $P$ is 1-1. Such
neighborhoods exist, because $P$ is 1-1 on the compact set $K_2$ and is
regular on a neighborhood of it. By Lemma 1 of [FS2], there is a domain
$B\subset V$ that is a neighborhood of $K_\frac{1}{16}$ and that is
biholomorphically equivalent to ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1$.

The Lemma is proved.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\begin{coro}
Let $U$ be a connected open set in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$. Suppose
$F_j:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j} \rightarrow U, j=1,2,\dots, N$ are finitely many
$1-1$ holomorphic maps
with $F_i({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_i}) \cap F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j}) =
\emptyset\; \forall\; i \neq j.$
Let $0<s<1$. Then there exists a $1-1$ holomorphic map
$F_0:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$ so that $F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)
\supset F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{sr_1})\cup_{j>1} F_j\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{sr_{j}}{32}}\right)
$
\end{coro}

{\bf Proof of the Corollary:}
We prove the Corollary by induction. By the Lemma, the result
holds for $N=2.$ Assume the result holds for some $N>1.$
We will show that the result holds for $N+1$.
Let
$F_j:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j} \rightarrow U, j=1,2,\dots, N+1$ be finitely many
$1-1$ holomorphic maps
with $F_i({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_i}) \cap F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j}) =
\emptyset\; \forall\; i \neq j.$
Let $0<s<1$. Pick a $\sigma, s<\sigma<1.$ Define $U_1:=
U \setminus \overline{F_{N+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_{N+1}})}.$ Then $U_1$
is a connected domain and
$F_j:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j} \rightarrow U_1, j=1,2,\dots, N$  are finitely many
$1-1$ holomorphic maps
with $F_i({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_i}) \cap F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{r_j}) =\emptyset\; \forall\; i \neq j
,i,j\leq N.$
Hence the inductive hypothesis applies to find a holomorphic map
$F_0:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1 \rightarrow U_1$ so that
$F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)
\supset F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_1})\cup_{1<j\leq N}
 F_j\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{\sigma r_j}{32}}\right)$
It follows that the two maps
$F_0:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$ and $F_{N+1}:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_{N+1}}\rightarrow U$
have disjoint range.
Next, let $\lambda, s<\lambda<\sigma$ be arbitrary and pick
$0<\tau<1$ large enough, then
$F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_\tau)\supset
 F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\lambda r_1})\cup_{1<j\leq N}
 F_j\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{\lambda r_{j}}{32}}\right).$

\medskip

Next we pick $\mu,0<\mu<1$ so large that
$$F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\mu\tau})\supset
 F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s r_1})\cup_{1<j\leq N} F_j\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{ s r_{j}}{32}}
\right).$$
We apply the Lemma to the maps $F_0:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_\tau\rightarrow U$ and $F_{N+1}:
{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r_{N+1}}\rightarrow U.$ Pick $s', 0<s'<1$ so that $\mu<s'$
and $s<s' \sigma.$ Then there exists by Lemma 2.1 a $1-1$ holomorphic
map $F:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$ so that

\begin{eqnarray*}
F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1) & \supset &
F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s'\tau})\cup F_{N+1}\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{s'\sigma r_{N+1}}{32}}
\right)\\
& \supset & F_0({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\mu\tau})\cup
F_{N+1}\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{s r_{N+1}}{32}}\right)\\
& \supset &
 F_1({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s r_1})\cup_{1<j\leq N} F_j\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{ s r_{j}}{32}}
\right)
\cup
F_{N+1}\left({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\frac{s r_{N+1}}{32}}\right)\\
\end{eqnarray*}

\hfill \vrule width 7pt height 7pt depth 1pt\\

Let $C$ denote a cube in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ and let $B$ denote a maximal
concentric ball in $C$. We let $\rho:=\rho_m:= \frac{|B|}{|C|}.$ Here
$|x|$ denotes the volume of $x$. Obviously $\rho_m$ is well-defined,
$i.e.$ independent of the size of $C$.

\begin{lem}
Let $V$ denote a bounded open set in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m.$ Then there exists a
finite number of disjoint balls $B_i \subset V$ so that
$|\cup B_i|> \frac{\rho|V|}{2}.$
\end{lem}

{\bf Proof:}
We can cover ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ by a grid of cubes so that the volume
of the collection of cubes entirely contained in $V$ has
volume at least $\frac{|V|}{2}.$ Then pick the largest concentric ball
in each of these cubes.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\begin{lem}
Let $F:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$ denote a $1-1$ holomorphic
map into a connected bounded open set $U\subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$.
Let $0<s<1$ and $\epsilon>0.$ Then there exists a $1-1$
holomorphic map
$F':{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$ so that
$F'({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)\supset F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_s)$ and $$|F'({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|
\geq |F(B^m_1)|+\frac{\rho}{2*(32)^{2m}}\left(|U|-|F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|\right)
-\epsilon.$$
\end{lem}

{\bf Proof:}
If $r, 0<s<r<1$ is large enough,
then the volume of $F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\setminus {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_r)$ is
at most $\epsilon/2.$
The set $V:U \setminus F(\overline{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_r})$ is connected.
By Lemma 2.3 we can find finitely many disjoint
balls $B_j$ in $V$ with total volume
at least $\frac{\rho |V|}{2}.$ Applying Corollary 2.2
to these balls $B_j^m(p_j,r_j)$ and the ball $F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_r)$ we find
a $1-1$ holomorphic image $F'({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)$ of the unit ball in $U$ which
contains $F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r})$ and $B^m_j(p_j, \frac{\sigma r_j}{32})$
for any $\sigma, 0<\sigma<1, s<\sigma r$ we want. By choosing $\sigma$
large enough,
we can suppose that the volume of $F'({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)$ is at least
\begin{eqnarray*}
|F'({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)| & \geq & |F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_r)|+ \frac{\rho |V|}{2*(32)^{2m}}-\epsilon/2\\
& \geq &
|F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|+ \frac{\rho |V|}{2*(32)^{2m}}-\epsilon\\
& \geq &
|F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|+ \frac{\rho (|U|-|F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|)}{2*(32)^{2m}})-\epsilon.\\
\end{eqnarray*}

\noindent By the choice of $\sigma,$ we also have that
$F'({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1) \supset F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{\sigma r}) \supset F({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_s).$\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\begin{coro}
Let $U$ be a bounded connected open set in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m.$ Then there exists
a sequence of $1-1$ holomorphic maps $F_j:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$
and numbers $0<s_j<1, j=1, \dots$ so that\\
\noindent (i) $F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j+1}}) \supset F_{j}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j}})\;
\forall\; j \geq 1$\\
\noindent (ii) $|F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j+1}})|
\geq |F_{j}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j}})|+\frac{\rho}{3*(32)^{2m}}\left(
|U|-|F_{j}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j}})|\right)\; \forall \; j \geq 1.$
\end{coro}

{\bf Proof:}
We prove the Corollary by induction. Let $F_1:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow U$
be an arbitrary ball, set $s_1=1/2.$ Suppose we have found
$F_1,\dots,F_j$ and $s_1,\dots,s_j.$ We apply Lemma 2.4 to
the map $F=F_j$ and $s=\frac{s_j+1}{2}>s_j$. Let
$\epsilon= \frac{\rho}{12*(32)^{2m}}\left(|U|-|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m
_{s_j})|\right).$
Then pick $F'$ as in Lemma 2.4 and set $F_{j+1}=F'$ Then
$$F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)\supset F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_s) \supset \supset F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})\;
{\mbox{and}}$$

\begin{eqnarray*}
|F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)| & \geq & |F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|+
\frac{\rho}{2*(32)^{2m}}\left(|U|-|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|\right)-\epsilon\\
& = &  |F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1)|(1-
\frac{\rho}{2*(32)^{2m}})+\frac{\rho}{2*(32)^{2m}}|U|-\epsilon\\
& \geq &
|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|(1-
\frac{\rho}{2*(32)^{2m}})+\frac{\rho}{2*(32)^{2m}}|U|-\epsilon\\
& = &
|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|+
\frac{\rho}{2*(32)^{2m}}\left(|U|-|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|\right)-\epsilon\\
& = &
|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|+
\frac{5\rho}{(12)*(32)^{2m}}\left(|U|-|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|\right)\\
& > & |F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|+
\frac{\rho}{3*(32)^{2m}}\left(|U|-|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|\right)\\
\end{eqnarray*}

Finally we choose $0<s_{j+1}<1$ large enough that

$$F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j+1}}) \supset F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})\;
{\mbox{and}}$$

$$|F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j+1}})|\geq |F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|+
\frac{\rho}{3*(32)^{2m}}\left(|U|-|F_j({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_j})|\right)$$

\hfill \vrule width 7pt height 7pt depth 1pt\\

\begin{prop}
Let $U$ be a bounded connected open set in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m.$ Then there exists
a sequence of $1-1$ holomorphic maps $F_j:{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_1\rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$
and numbers $0<s_j<1, j=1, \dots$ so that\\
\noindent (i) $F_{j+1}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j+1}}) \supset F_{j}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j}})\;
\forall\; j \geq 1$\\
\noindent (ii) $|F_{j}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j}})| \rightarrow |U|$
\end{prop}

{\bf Proof:}
Let $F_j$ and $s_j$ be as in the Corollary 2.5.
Set $A:=\lim_j |F_{j}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}^m_{s_{j}})|.$ From (ii) it follows
that

$$
|U| \geq A \geq A+\frac{\rho}{3*(32)^{2m}}(|U|-A).
$$

This is impossible if $A<|U|.$\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

{\bf Completion of the Proof of Theorem 1.4:}
The union $\cup_{j} F_j(B^m_{s_j})$ is biholomorphic to the unit ball.
This follows from [FS1] since the union is contained in a
Kobayashi hyperbolic set.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\section{Proof of Theorem 1.3}

We prove first Theorem 1.5.

{\bf Proof:}
Let $c$ be a Lipschitz constant of $\phi.$ Then

$$
\frac{|\phi(v,z)-\phi(v_0,z_0)|}{\|(v,z)-(v_0,z_0)\|}
\leq c.$$

 \noindent This is equivalent to the following:
Let $\Gamma=\{u=\phi(v,z)\}$ and choose a point $(u_0+iv_0,z_0)\in \Gamma,$
then $\Gamma$ does not intersect the cone

$$
{\mathcal C}_{(z_0,u_0,v_0)}:=\{(z,u+iv);|u-u_0|> c \|(z,v)-(z_0,v_0)\|.
$$

\noindent in the point $(u_0+iv_0,z_0).$ Moreover let $(\alpha,\beta)
\in {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^{m-1}\times \RR$ be such that
$\|(\alpha,\beta)\|<\frac{1}{c}$, then all points
$(z_0,u_0+iv_0)-t(\alpha,1+i\beta)$ where $0<t$ are contained
in $\Omega.$ Let

$$
 H_{t,\alpha,\beta}(z,u+iv)=(z,u+iv)-t(\alpha,1+i\beta).
$$

\noindent For fixed $t,\alpha$ and $\beta$ this is a holomorphic map and

$$
(*)\;\;\;\;\;\;\;\;\;\;\;\;\; H_{t,\alpha,\beta}(\Omega) \subset \Omega.
$$

\bigskip

We want to show that $\tilde{\Omega}$ also has this property $(*)$.
Since $\Omega$ is bounded by a graph it follows that $\tilde{\Omega}$
is schlicht and also is bounded by a graph (see [S]). In fact,
$\tilde{\Omega}=\cup_{j=0}^\infty \Omega_j, \Omega_0=\Omega,$ where $\Omega_j$
consists of all the points that can be reached from $\Omega_{j-1}$
by pushing discs with boundaries in $\Omega_{j-1}$. We make this more
precise: Let $H$ denote the Hartogs figure

$$
H:=\{p\in
\overline{\Delta}_0 \cup \cup_{0\leq s\leq 1}\partial \Delta_s\},\;
\Delta_s=\{(z,t)\in {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C} \times \RR; t=s, |z|\leq 1\}\subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^2.
$$

Moreover, $\hat{H}$ denotes the convex hull of $H.$ Let $\Phi$ denote
any rank $2$, one-to-one holomorphic map defined on some (not fixed)
neighborhood of $\hat{H}$ in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^2$ with values in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m.$ Then

$$
\Omega_j:= \cup_{\{\Phi; \Phi(H)\subset \Omega_{j-1}\}}
\Phi(\hat{H}).
$$

\medskip

Now we only need to show that if $\Omega_{j-1}$ has property $(*)$
so does $\Omega_{j}.$

\medskip

Observe that if $\Phi(H)\subset \Omega_{j-1}$, then
$H_{t,\alpha,\beta}(\Phi(H))= (H_{t,\alpha,\beta}\circ \Phi)(H))
\subset \Omega_{j-1}.$ Hence if $p\in \Omega_j,$
then $H_{t,\alpha,\beta}(p)\in \Omega_j.$

\bigskip

Now $\partial \tilde\Omega =\tilde{\Gamma}$ where

$$
\tilde{\Gamma}=\{(z,u+iv); u=\tilde{\phi}(z,v)\}.
$$

Obviously, by $(*)$, if $\tilde{\phi}(z,v)$ is infinite for
some $(z,v),$ then $\tilde{\phi} \equiv \infty$ and we are done.
We assume hence that $\tilde{\phi}$ has values in $\RR$ only.

We need to show that if $(z_0,u_0+iv_0)\in \tilde{\Gamma},$ then
$\tilde{\Gamma}$ does not intersect the cone ${\mathcal C}_{(z_0,u_0,v_0)}.$

We know that the cone does not meet $\tilde{\Gamma}$
in points $(z,u+iv)$ where $u<u_0$ since these are
interior points of $\tilde{\Omega}.$

\medskip

Assume that $(z_1,u_1+iv_1)\in \tilde{\Gamma}\cap {\mathcal C}_{(z_0,u_0,v_0)}$
and $u_1>u_0.$ We want to show that this implies that
$(z_0,u_0+iv_0)\in \tilde{\Omega}$ which contradicts the fact that this is a
boundary point. But the fact that $(z_1,u_1+iv_1)\in \partial \tilde{\Omega}$
implies that the set

$$
\{(z,u+iv)=(z_1,u_1+iv_1)-t(1,\alpha,\beta); \|(\alpha,\beta)\|
<\frac{1}{c},0<t<1\}
$$

\noindent is contained in $\tilde{\Omega}$ and a simple computation
shows that $(z_0,u_0+iv_0)$ is an interior point of this set.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

{\bf Proof of Theorem 1.3:}
Pick a point $p\in \partial \Omega$. If $p$ is an interior point
of the envelope of holomorphy then we are done. So suppose not.
Since $ \Omega$ has Lipschitz boundary, we can rotate and translate coordinates
so that $p=0$ and $\Omega$ is locally given by
$\Omega=\{(z,u+iv);u<\phi(z,v),\|(z,v)\|<\rho\}$ for some Lipshitz function $\phi.$
let $A>\phi(0)$ and pick $L>0$ large enough that
$A-L\|(z,v)\|<\phi(z,v)$ if
$\rho/2<\|(z,v)\|<\rho.$ We globalize $\Omega$ by setting

\begin{eqnarray*}
\Omega'  & = & \{(z,u+iv)\in {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m; u-L\|(z,v)\|\; {\mbox{if}}\;
\|(z,v)\|>\rho/2,\\
& &
u<\min\{A-L\|(z,v)\|,\phi(z,v)\}\;{\mbox{if}}\;
\|(z,v)\|\leq \rho/2\}.
\end{eqnarray*}

Letting $\tilde{\Omega}'$ be the envelope of holomorphy
of $\Omega'$, $\tilde{\Omega}'=\{u<\tilde{\phi}(z,v)\},$
we get $\tilde{\phi}(0)=\phi(0)=0,
\tilde{\phi}\geq \phi$ near $0.$
By a
theorem by Demailly ([De]) it follows that there is a
locally defined continuous function $\rho$ on the set
$\{u\leq \tilde{\phi}(z,v)\}$ which is plurisubharmonic on
 $\{u< \phi(z,v)\} \subset\{u< \tilde{\phi}(z,v)\}$
and with $\rho(0)=0.$\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\section{Proof of Theorem 1.2}

We prove the theorem in steps.

\bigskip

Step 1:\\

Let $R'$ be the closed rectangle in ${\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}$ with corners
$$(0,0), (2\sqrt{2}\pi,0), (0,4\pi), (2\sqrt{2}\pi,4\pi).$$ This is covered
by two closed rectangles $R_1',R_2'$ where $R_1'$ has corners
$$(0,0), (2\sqrt{2}\pi,0), (0,2\pi), (2\sqrt{2}\pi,2\pi)$$
\noindent and $R'_2$
has corners  $(0,2\pi), (2\sqrt{2}\pi,2\pi), (0,4\pi), (2\sqrt{2}\pi,4\pi).$

Let $F_1:R_1'\rightarrow R', F_1(z)=\sqrt{2}i z+2\sqrt{2}\pi$
and $F_2:R_2'\rightarrow R', F_2(z)=\sqrt{2}iz+4\sqrt{2}\pi.$ Then
the $F_j$ are homeomorphisms.
Next let $R$, $R_j$ denote the interiors of the same rectangles.
Let $\Omega:=R_1\cup R_2$ and set $F:\Omega \rightarrow R, F(z)=
F_1(z), z\in R_1, F(z)=F_2(z), z\in R_2.$
Let $p$ be any point in $\Omega$ and let $U\subset \Omega$ be some
neighborhood of $p.$ Then there exists a connected
open subset $V\subset U$ and an
integer $N\geq 1$ so that $F^j(V)\subset \Omega$ for each $1\leq j
\leq N,F^N(V)=R_1$ or $R_2$ and hence $\Omega \subset F^{N+1}(V).$
From this observation it is easy to inductively find a dense
orbit $\{p_n\}$ in $\Omega.$ Moreover, we can choose $\{p_n\}$
to be contained in any open subset of full measure or just dense.\\

Step 2:\\

Let $D$ denote a disc,

$$
D:=\{w\in {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}; 1<|w|<e^{2\sqrt{2}\pi},w\notin \RR^+\setminus {0}\}.
$$

Then the maps $\tau_j:R_j \rightarrow D,
\tau_j(z)=e^z$ are biholomorphisms, $j=1,2.$
Suppose that $z_j\in R_j, z_2=z_1+2\pi i.$ Then $\tau_1(z_1)=
\tau_2(z_2).$
Notice also that
\begin{eqnarray*}
F_2(z_2) & = & \sqrt{2}iz_2+4\sqrt{2}\pi\\
& = & \sqrt{2}iz_1-2\sqrt{2}\pi+4\sqrt{2}\pi\\
& = & F_1(z_1)\\
\end{eqnarray*}

Hence, we can define $\Phi:D \rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}$ by $\Phi(w)=e^{F_j(z_j)},
j=1\; \mbox{or}\; 2,w=e^{z_j},z_j\in R_j.$
Then $\Phi$ is a well defined holomorphic map. Let $\{p_n\}_{n=0}^\infty$
be a dense orbit of $F$ in $\Omega.$ Define $q_n:=e^{p_n}\in D.$
Then $\Phi(q_n)=e^{F_j(p_n)}$ if $p_n\in R_j.$
Hence $\Phi(q_n)=e^{F(p_n)}=e^{p_{n+1}}=q_{n+1}.$ Hence
$\{q_n\}\subset D$ and is a dense orbit for $\Phi.$ \\

Step 3: \\

Let $D^m:= D \times \cdots \times D$, $m$ times and define
$\Psi:D^m \rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$ by
$\Psi(z_1,\dots,z_m)= (\Phi(z_1),\dots, \Phi(z_m)).$
Then we can find a dense sequence $\{p_n\}\subset D^m$ contained
in any given dense open subset.\\

Step 4:\\

To finish the proof of Theorem 1.2 we use Theorem 1.4 to find
an open subset of $D^m$ of full measure and biholomorphic
to the unit ball.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\section{Proof of the Main Theorem}

To prove the Theorem, we assume to the contrary that there are such $V$ and
$\{p_n\}.$ We can assume $V$ is connected and $(0)\in V.$
We let $\tilde{\Omega}$ denote the (possibly multisheeted)
envelope of holomorphy of $\Omega$ and $\Omega^*\subset
\tilde{\Omega}$ consists of $\Omega \cup \{q\in \partial \Omega;
q\in \tilde{\Omega}\}.$ The map $F$ extends holomorphically
to a map $\tilde{F}:\tilde{\Omega} \rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m.$ We denote by
$F^*$ the restriction of $\tilde{F}$ to $\Omega^*.$ So $F^*$ is a continuous
extension of $F$ to $\Omega^*$ and in turn $F^*$ extends holomorphically.
We define $\Omega_k:=\{z=z_0\in \Omega; z_1:=F^*(z)\in \Omega^*,
z_2:=F^*(z_1)\in \Omega^*,\dots, z_k:=F^*(z_{k-1})\in \Omega^*\}.$
Note that in general the $\Omega_k$ are not open sets but
the maps $(\tilde{F})^{k+1}$ are well defined holomorphic maps in small
neighborhoods of the $\Omega_k.$

\begin{lem}
We have that $V \subset \Omega_k$ for all $k.$
\end{lem}

{\bf Proof of the Lemma:}
We show first that $F(V) \subset \Omega_1.$
If $p_k\in V$, then $F(p_k)=p_{k+1}\in \Omega.$ Since
$\{p_n\}\cap V$ is dense in $V$, it follows by continuity
that $F(V) \subset \overline{\Omega}.$ Suppose there is a point
$q=F(p)\in \partial \Omega,p\in V.$ We will show that $q\in \Omega^*.$
If $q \notin\Omega^*$, we can find, by Theorem 1.3, an open
neighborhood $U(q)$ and a continuous function $\rho :U\cap \overline{\Omega}
\rightarrow (-\infty,0]$ which is plurisubharmonic on $U \cap \Omega$,
$\rho<0$ on $U \cap \Omega$ and $\rho(q)=0.$ Since $\Omega$
has Lipschitz boundary, we can find an outward vector $\bf{n}$ in a possibly
smaller neighborhood of $q.$
Let $\Phi_\epsilon, \epsilon>0$ denote the inward translation
by $\epsilon$ in the direction of $\bf n.$ Then there exists
a fixed neighborhood $q\in W\subset V$ so that
$\rho\circ \Phi_\epsilon\circ F$ is plurisubharmonic on
$W$ for all small enough $\epsilon$ and moreover
$\rho \circ \Phi_\epsilon \circ F \rightarrow \rho\circ F$
uniformly as $\epsilon \searrow 0.$ It follows that
$\rho\circ F$ is plurisubharmonic on $W$ and since this
function takes its maximum at $q$ it must be constant on $W.$
This is a contradiction since $\rho\circ F<0$ at each point
of $\{p_n\}\cap U.$ This shows that $V \subset \Omega_1.$ \\

Suppose we have
shown that $V \subset \Omega_k.$ Then $F^{k+1}$ extends holomorphically
to $V$. The same argument as above shows that the image
of the extension is contained in $\overline{\Omega}.$ Furthermore,
applying theorem 1.3 again in the same way as above, we get that
the image is actually contained in $\Omega^*.$ This shows that
$V \subset \Omega_{k+1}$, completing the induction.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

{\bf Proof of the Theorem:}\\

Since $(0)\in V,$ we can
choose sequences $m(k)<\ell(k)$ so that $p_{m(k)}, p_{\ell(k)}
\rightarrow 0.$ We set $n(k)=\ell(k)-m(k)$. We can assume that
$n(k) \nearrow \infty$. By the above Lemma, the map $F^{n(k)}$ has a
holomorphic extension to $V$ and $F^{n(k)}(p_{m(k)})=p_{\ell(k)}.$
Restricting to a thinner subsequence if necessary,
we can assume that $F^{n(k)} \rightarrow G: V \rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m,$
$G(0)=0.$

\medskip

For any positive integer $\tau$ the sequence
$F^{\tau n(k)}=({F^{n(k)}})^{\tau}$ converges in a small
enough neighborhood to $G^\tau$ and by a normal families
argument, this convergence extends to all of $V$ and a suitable
extension of $G^\tau$ to $V$. The image of $V$ is still
contained in $\overline{\Omega}.$

\bigskip

We get:

\begin{lem}
Suppose that $F^{n_k}:V \rightarrow \overline{\Omega}$ is any sequence of
iterates converging to some map $G:V \rightarrow \Omega, G(0)=0,$
then all iterates $G^\tau$ can also be analytically continued
from $0$ to a map from $V$ to $\overline{\Omega}$. Moreover, if
$G^{\tau_s}$ is any convergent subsequence to a map
$H: V\rightarrow {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}^m$, then there is a subsequence
$F^{m(s)}$ converging to $H$ on $V$, moreover,
$H(0)=0, H(V) \subset \overline{\Omega}.$
\end{lem}

Using this Lemma we can refine the properties of
a limit $G$ of some subsequence $F^{n_k}$.\\

Let $\{\lambda_j\}_{j=1}^m$ denote the eigenvalues
of $G'(0)$. Since $G^\tau(V) \subset \overline{\Omega}$ for all
$\tau$, it follows that all eigenvalues have modulus $\leq 1.$
We can assume that
$1\geq |\lambda_1| \geq \cdots \geq |\lambda_m|.$
Using the Lemma again, we can assume that $\lambda_i=1,
i\leq j, \lambda_i=0, i>j$ for some $0 \leq j\leq m.$
If we put the matrix $G'(0)$ in Jordan normal form, we see that
for the blocks with eigenvalue $1$ there can be no off diagonal term
because these would grow under iteration violating Schwarz's Lemma.
Also if we look at the off diagonal terms for blocks with eigenvalue
zero, these vanish after iteration, so hence we can assume that there are
no off-diagonal terms.

\bigskip

Case (i): $\lambda_1=\cdots =\lambda_m=0.$\\

Then there exists $0<s<r$ and $k$ sufficiently large that
$F^{n_k}({\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,r))\subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,s)\subset {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,r)\subset V$ and
$\|(F^{n_k})'(a)(v)\|\leq c\|v\|$ for some $c<1$ and
for all $a\in {\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}(0,r).$
Hence $F^{n_k}$ is a contraction and hence has an attracting fixed
point. Therefore $F$ has an attracting periodic orbit.
This is impossible since there is a dense orbit.\\

Case (ii): $\lambda_1= \cdots =\lambda_m=1.$\\

In this case $G'(0)=\mbox{Id}$. If $G$ has nonzero higher order
terms in the Taylor expansion around the origin, the iterates of $G$
cannot be a normal family on $V$, so $G(z)\equiv z$ on $V$.

\medskip

Let $\hat{V}:=V \cup \cup_{n\geq 1}{\tilde{F}}^n(V).$ Then $\hat{V}$
is open and $\tilde{F}(\hat{V})\subset \hat{V}.$

\begin{lem}
The set $\hat{V}$ has finitely many connected components
$U_1,\dots,U_r=U_1$ with $\tilde{F}(U_i)\subset U_{i+1}.$ In fact
the map $\tilde{F}:U_i\rightarrow U_{i+1}$ is a biholomorphism and
${\tilde{F}}^{n_k} \rightarrow {\mbox{Id}}$ on each $U_i.$
\end{lem}

{\bf Proof of the Lemma:}
Clearly $\tilde{F}^m$ must be $1-1$ on $V$ for each iterate $\tilde{F}^m$.
Hence the Jacobians of each iterate is everywhere nonzero and
each iterate is an open map. Hence $\hat{V}$ is an open set.
Moreover, $\tilde{F}(\hat{V})\subset \hat{V}$. In particular,
each connected component is mapped into another connected component
by $\tilde{F}$. Let $U_1$ be the connected component containing $V$ and
define inductively connected components $U_i$ by $\tilde{F}
(U_i)\subset U_{i+1}$.
Since $\tilde{F}^{n_k}\rightarrow {\mbox{Id}}$ on $V$ it follows that for a minimal
$r, U_r=U_1.$ Hence there are only finitely many connected components.
The condition that $\tilde{F}^{n_k}\rightarrow {\mbox{Id}}$ on $V$ implies
that $F:\hat{V} \rightarrow \hat{V}$ is onto.
Hence each map $\tilde{F}:U_i\rightarrow U_{i+1}$ is onto.
By the identity theorem
it follows that $\tilde{F}^{n_k}\rightarrow {\mbox{Id}} $ on each $U_i$. Hence
 $\tilde{F}:U_i\rightarrow U_{i+1}$ is $1-1$ and hence a biholomorphism.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

This implies that ${\tilde{F}}^r:U_1 \rightarrow U_1$ is an automorphism
with a dense orbit of a bounded domain. The automorphism group is a
Lie group and the closure of the set of iterates of ${\tilde{F}}^r$
is a commutative
closed subgroup, isomorphic to a product of a torus and Euclidean
space and a finite commutative group.
Since there is a sequence of iterates of ${\tilde{F}}^r$
converging to the identity, the Euclidean component must vanish,
but then the orbit
of any point is nowhere dense, a contradiction.\\

Case (iii): $\lambda_1= \cdots =\lambda_j=1, \lambda_{j+1}=\cdots =
\lambda_m=0$
for some $1 \leq j <m.$
Then $G$ is of the form
$$
(**)\;\;\;\;\;\;\;
G(z_1,\dots,z_m)=(z_1+f_1,\dots,z_j+f_j,f_{j+1},\dots,f_m), f_i=
{\mathcal O}(\|z\|^2)$$

Fix a small neighborhood $W, 0\in W \subset \subset U_1$.
Let $U \subset \subset W$ and suppose that ${\tilde{F}}^r(U) \subset W.$
Then we have the
functional equation $${\tilde{F}}^r\circ G(z)=G \circ {\tilde{F}}^r(z),
z\in W.$$

Let $z_\delta:=(0,\c\dots,0,\delta)$ for small enough $\delta \neq 0.$
Using the density of orbits and equicontinuity there exists for any
$\epsilon>0$ a neighborhood $U(0)\subset W$ and an integer $r_\epsilon>1$
so that $\|{\tilde{F}}^{r_\epsilon}(0)-z_\delta\|\leq \epsilon$ and
${\tilde{F}}^r(U)\subset W.$
Since $G(0)=0$, ${\tilde{F}}^{r_\epsilon}(0)=G({\tilde{F}}^{r_\epsilon}(0))$.
This is
clearly impossible by (**). This completes the proof.\\

\hfill \vrule width 7pt height 7pt depth 1pt\\

\begin{remark}
We can relax the Lipschitz condition of The Main Theorem if we add more
hypotheses on $F:$ Let $\Omega$ be any domain
with $\Omega={\mbox{int}}(\overline{\Omega})$
and assume in addition that the map $F$ has discrete fibers.
Then $F$ has no dense orbit.
\end{remark}


\section{INTRODUCTION}
The Tevatron Collider Run II started in March 2002 and is expected
to continue until the end of this decade. The Tevatron and the
two detectors, CDF and D\O, have been performing  well in 2004,
each experiment is collecting data at the rate
of $\approx$10 pb$^{-1}$ per week.
The total  luminosity accumulated by August 2004 is $\approx$500 pb$^{-1}$
per detector.
The rich physics program includes the
production and precision measurement of properties of  standard model (SM)
objects, as well as searches for phenomena beyond standard model.
In this brief review we focus on areas of most interest
to the lattice community. We present
new results on the top quark mass
and their implication for the mass of the SM Higgs boson,
on searches for the SM Higgs boson, on evidence for the $X(3872)$ state,
on searches for pentaquarks, and on $b$ hadron properties.
All Run II results presented here are preliminary.

\section{TOP QUARK MASS}

The experiments CDF and D\O\ published several direct  measurements of
the top quark pole mass, $\ensuremath{M_{\mathrm{top}}}$,
based on Run I data (1992-1996).
The ``lepton $+$ jets'' channel yields the most precise determination of
$\ensuremath{M_{\mathrm{top}}}$. Recently, the
D\O\ collaboration published a new measurement~\cite{Mtop1-D0-l+j-new},
based on a powerful analysis technique yielding  greatly improved precision.
The differential probability
that the measured variables in any event correspond to the signal
is calculated as a function of $\ensuremath{M_{\mathrm{top}}}$.
The maximum in the product of the individual event probabilities
provides the best estimate of $\ensuremath{M_{\mathrm{top}}}$.
The critical differences from previous analyses
in the lepton $+$ jets decay channel lie in
the assignment of more
weight to events that are well measured or more likely to correspond to
$t \bar t$ signal,
and  the handling of the combinations of final-state objects
(lepton, jets, and imbalance in transverse momentum)
and their identification with
top-quark decay products in an event.
The new combined value for the top-quark mass from Run I is
$\ensuremath{M_{\mathrm{top}}}  =  178.0\pm4.3~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.

In Run II, both collaborations  have been exploring several different techniques
for $\ensuremath{M_{\mathrm{top}}}$
measurements. The best single CDF result comes from a dynamic likelihood method
(DLM). The method is similar to
the technique used in Ref.~\cite{Mtop1-D0-l+j-new}.
The result is $\ensuremath{M_{\mathrm{top}}} = 177.8^{+4.5}_{-5.0} (stat) \pm  6.2 (syst) ~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.
The joint likelihood of the selected events is shown in Fig. ~\ref{fig:cdf_tml}.
The Run II goal is a 1\% uncertainty on $\ensuremath{M_{\mathrm{top}}}$.

\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=5.8cm,width=8.1cm]  {data_22ev_likelihood.eps}
\vspace*{-1.2cm}
\caption{The joint likelihood of top candidates(CDF).}
\label{fig:cdf_tml}
\end{figure}

\section{SEARCH FOR SM HIGGS BOSON}

The constraints on the SM Higgs ($H$)  boson  mass from
published  measurements, updated to include the new D\O\ top mass
measurement~\cite{Mtop1-D0-l+j-new}, are
$M_H = 117 ^{+67}_{-45}~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$, $M_H < 251~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$ at 95\% C.L.
The  new most likely  value of $M_H$
is above the experimentally excluded range,
and sufficiently low for $H$ to be observed at the Tevatron.

\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=7.5cm,width=7.8cm]  {d0_wbb_fig_3_err.eps}
\vspace*{-1.1cm}
\caption{Distribution of the dijet
invariant mass for $W+2 b$-tagged jets  events,
compared to the expectation (D\O).
}
\label{fig:d0_wbb_2tag}
\end{figure}

D\O\  has conducted a search for $H$ at $M_H < 140~\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$
in the production channel
$p \bar{p} \rightarrow WH \rightarrow  e \nu b \bar{b}$.
The experimental signature of  $WH \rightarrow e \nu b \bar{b}$
is a final state with
one high $p_T$ electron, two  $b$ jets, and
large missing transverse energy  resulting from
the undetected neutrino.
The dominant backgrounds to $WH$ production
are  $W b \bar{b}$, $t \bar{t}$ and single-top production.
The distribution
of the dijet mass for events with two $b$-tagged jets is shown in
Fig.~\ref{fig:d0_wbb_2tag}.
Also shown is the  expected contribution ($0.06$ events)
from the $b \bar{b}$ decay of a
SM Higgs boson with $M_H =$ 115 $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.
No events are observed in the  dijet mass window of 85--135  $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$.
D\O\ sets a limit on the cross section
for $\sigma( p\bar{p} \rightarrow WH) \times B(H \rightarrow b \bar{b}) $
of 9.0 pb at the 95\% C.L.,  for a 115  $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$ Higgs boson.
The results for mass points 105, 125, and 135 $\ensuremath{\mathrm{ Ge\kern -0.1em V }\kern -0.2em /c^2 }$
 are 11.0, 9.1 and 12.2 pb,  respectively.

\begin{figure}[htb]
\vspace*{-1.2cm}
\includegraphics[height=0.33\textheight,width=8.0cm]{whww_aps04_bw.eps}

\vspace*{-1.2cm}
\caption{95\% limits on the $H$ production (CDF).}
\label{fig:cdf_whww}
\end{figure}

CDF  has done  a similar search, allowing either an  electron or a muon
in the final state.  Both groups have also searched for $H$ produced in
gluon-gluon fusion, with subsequent decay to a pair of $W$ bosons.
The CDF results for both channels  are shown in Fig.~\ref{fig:cdf_whww}.

\section{THE STATE X(3872)}

\begin{figure}[htb]

\includegraphics[height=8.0cm,width=7.5cm]  {X3872cdfPRL1FullM.eps}
\vspace*{-1cm}
\caption{The $X(3872)$ signal (CDF).}
\label{fig:cdf_x}
\end{figure}

 The existence of the $X(3872)$ state discovered by
the Belle Collaboration~\cite{Belle-X}
 has been confirmed
 in $p \bar{p}$ collisions by  CDF~\cite{cdf-X} (see Fig.~\ref{fig:cdf_x})
and D\O~\cite{d0-X}.
 It is still unclear whether this particle is a $c\bar{c}$ state,
 or a more complex object.  When the data are separated according to
production and decay variables, D\O\  finds no significant
differences between the $X(3872)$ and
the $c \bar{c}$ state $\psi(2S)$.
CDF has analysed the ``lifetime'' distribution of the $X(3872)$ events in order to
quantify what fraction of this state arises from decay of $B$ hadrons, as opposed to
those produced promptly. The authors find that for the selected samples
28.3$\pm$1.0$(stat)\pm$0.7$(syst)$\% of $\psi(2S)$ candidates are from $b$ decays,
whereas 16.1$\pm$4.9$(stat)\pm$2.0$(syst)$\% of $X$ mesons arise from such decays.

\section{SEARCH FOR PENTAQUARKS}

\begin{figure}[htb]

\includegraphics[height=0.27\textheight,width=7.6cm]  {mpks_1stminbias.eps}
\vspace*{-1.2cm}

\caption{Invariant mass distribution of an identified proton and a $K^0_s$ candidate. (CDF)
}
\label{fig:pqtheta}
\end{figure}

\begin{figure}[htb]

\vspace*{-0.9cm}
\includegraphics[height=0.25\textheight,width=8.0cm]  {CM_xicst_cc_1.eps}
\vspace*{-1.2cm}
\caption{Invariant mass distribution of the $(\Xi^-,\pi^+)$ system. (CDF)
}
\label{fig:pqxi}
\end{figure}

\begin{figure}[htb]
\vspace*{-0.9cm}

\includegraphics[height=0.25\textheight,width=7.6cm]  {theta_note_dstp_dedx_pt.eps}
\vspace*{-1.2cm}
\caption{Mass of the ($D^{*+}\bar p$) system. The arrow indicates the position of
the $\Theta_c$ state (CDF).}
\label{fig:pqthetac}
\end{figure}

Following reports of evidence for exotic
baryons containing five quarks (pentaquarks), CDF has analysed
its data for evidence of the following pentaquarks:
$\Theta^+$ ($uud\bar d \bar s$), doubly strange states
$\Xi_{3/2}$, charmed states $\Theta_c$, and, most recently,
a state $(udus\bar b)$, dubbed $R^+_s$, through its weak decay to $(J/\psi, p)$.
With its excellent particle indentification and mass resolution,
CDF has a unique capability to search for  pentaquark states.
The signals of known states: $\phi$, $\Lambda$,
$\Lambda(1520)$, $K^*$, $\Xi$,
compare favorably with those provided
by the authors of  the pentaquark evidence.
The group finds no evidence for pentaquark states, see Figs
~\ref{fig:pqtheta},{\ref{fig:pqxi},\ref{fig:pqthetac}.
This can be interpreted as an indication that the pentaquark production
in $p \bar p$ collisions is heavily suppressed compared to the conventional
hadron production, or as an evidence against the existence of pentaquarks.

\clearpage

\section{RECENT B PHYSICS RESULTS}

\subsection{Spectroscopy}

CDF has measured the mass of $b$ hadrons in exclusive $J/\psi$ channels.
The measurements of the $B_s$ and $\Lambda_b$ (Fig. \ref{fig:masslb})
masses are the current world's best.\\

$m(B^+)$ = 5279.10$\pm$0.41$(stat)\pm$0.36$(syst)$,

$m(B^0)$ = 5279.63$\pm$0.53$(stat)\pm$0.33$(syst)$,

$m(B_s)$ = 5366.01$\pm$0.73$(stat)\pm$0.33$(syst)$,

$m(\Lambda_b)$ = 5619.7$\pm$1.2$(stat)\pm$1.2$(syst)$ MeV/$c^2$.\\

\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0.30\textheight,width=7.5cm]  {lambdav1c.eps}
\vspace*{-1cm}

\caption{The mass spectrum of $\Lambda_b$ candidates (CDF).}
\label{fig:masslb}
\end{figure}

D\O\ reports the first observation of the excited $B$ mesons
$B_1$ and $B^*_2$ as two separate states in fully reconstructed
decays to $B^{(*)}\pi$. The mass of $B_1$ is measured to be
5724$\pm$4$\pm$7 MeV/c$^2$, and the mass difference $\Delta M$ between
$B^*_2$ and $B_1$ is 23.6$\pm$7.7$\pm$3.9 MeV/c$^2$
(Fig.  \ref{fig:d0_bexc}).

D\O\ observes semileptonic $B$ decays to narrow $D^{**}$ states,
the orbitally excited states  of the $D$ meson
seen as resonances in the $D^{*+}\pi^-$ invariant mass spectrum.
The $D^*$ mesons are reconstructed through the decay sequence
$D^{*+} \rightarrow D^0\pi^+$, $D^0\rightarrow K^-\pi^+$.
The invariant mass  of oppositely charged $(D^*,\pi)$ pairs
is shown in Fig.  \ref{fig:d0_dstst}.
The mass peak between 2.4 and 2.5 GeV/$c^2$ can be interpreted as two merged
narrow $D^{**}$ states, $D^0_1(2420)$ and $D^0_2(2460)$.
The combined branching fraction is
$ {\cal B}(B\rightarrow D^0_1,D^0_2)\cdot {\cal B}(D^0_1,D^0_2\rightarrow D^{*+}\pi^-)=(0.280\pm0.021(stat)\pm0.088(syst)$\%. The systematic error includes the unknown phase between the
two resonances. Work is in progress on extracting the two Breit-Wigner
amplitudes.

\begin{figure}[htb]
\vspace*{-2mm}
\hspace*{-3mm}
\includegraphics[height=0.28\textheight,width=8.3cm]  {B08F02.eps}

\vspace*{-1cm}
\caption{Mass difference $\Delta M = M(B\pi)-M(B)$ for exclusive $B$ decays.
The background-subtracted signal is a sum of
$B^*_1 \rightarrow B^* \pi$, $B^* \rightarrow B \gamma $ (open area)
and $B^*_2 \rightarrow B^*\pi$ $B^*\rightarrow B \gamma$ (lower peak in the shaded area)
and $B^*_2 \rightarrow B \pi$ (upper peak in the shaded area)
(D\O).}
\label{fig:d0_bexc}
\end{figure}

\begin{figure}[htb]
\includegraphics[height=0.25\textheight,width=7.5cm]  {B05F03.eps}

\vspace*{-1cm}
\caption{The invariant mass distribution of
$(D^*,\pi)$ pairs, opposite sign (points) and same-sign (solid histogram).}
\label{fig:d0_dstst}
\end{figure}

\subsection{Lifetimes}

CDF and D\O\ have measured  lifetimes of $b$ hadrons through the exclusively
reconstructed decays $B^+ \rightarrow J/\psi K^+$, $B^0 \rightarrow J/\psi K^{*0}$,
$B_s \rightarrow J/\psi \phi$,
and $\Lambda_b \rightarrow J/\psi \Lambda$
(Fig. \ref{fig:d0_lbctau}).
The latest results are:  \\

 $\tau(B^+)$=1.65 $\pm$ 0.08 $^{+0.096}_{-0.123}$  ps ~(D\O\ 2003),

 $\tau(B^+)$=1.662 $\pm$ 0.033  $\pm$ 0.008  ps ~(CDF),

 $\tau(B^0_d)$=1.473  $^{+0.052}_{-0.050}$ $\pm$ 0.023    ps ~(D\O).

 $\tau(B^0_d)$=1.539 $\pm$ 0.051  $\pm$ 0.008  ps ~(CDF),

 $\tau(B^0_s)$=1.444   $^{+0.098}_{-0.090}$ $\pm$ 0.020   ps ~(D\O),

 $\tau(B^0_s)$=1.369 $\pm$ 0.100 $\pm$ $^{+0.008}_{0.010}$  ps ~(CDF),

 $\tau(\Lambda_b)$=1.221 $^{+0.217}_{-0.179}$ $\pm$ 0.043  ps ~(D\O),

 $\tau(\Lambda_b)$=1.25 $\pm$ 0.26 $\pm$ 0.10  ps ~(CDF 2003).\\

The measured lifetimes correspond to the following lifetime ratios:\\

$\tau(B^+)/\tau(B^0_d)$   =  1.080$\pm$0.042     ~(CDF),

$\tau(B^0_s)/\tau(B^0_d)$ =  0.890$\pm$0.072    ~(CDF),

$\tau(B^0_s)/\tau(B^0_d)$ = 0.980$ ^{+0.075}_{-0.070}   \pm$0.003    ~(D\O),

$\tau(\Lambda_b)/\tau(B^0_d)$ =  0.874$ ^{+0.169}_{-0.142}   \pm$0.028    ~(D\O).\\

\begin{figure}[htb]
\includegraphics[height=0.3\textheight,width=8.2cm]  {d0_lbctau_B11F02.eps}
\vspace*{-1cm}

\caption{ Fit projection on  $c\tau$ for the $\Lambda_b$ candidates.  (D\O)}
\label{fig:d0_lbctau}
\end{figure}

The $B_s$ lifetime measurements listed above are results of
a single-lifetime fit to data, integrated over the decay angles.
Because  of the presence of  final
states  common to ${B_s^0}$\ and its charge conjugate ${\overline{B}_s^0}$,
the two meson states   are expected
to mix in such a way that the two CP  eigenstates may have a relatively
large lifetime difference.
It is possible to
separate the two CP components of ${B_s^0 \rightarrow J/\psi \phi}$\ and thus to measure the
lifetime difference by studying the time evolution of the
polarization states of the vector mesons in the final state.
CDF has carried out a combined analysis of $B_s$ lifetimes
and polarization amplitudes. The results for the lifetimes of the
low mass (CP even) and high mass (CP odd) eigenstates, and the relative
width difference are:\\

 $\tau_L = 1.05 ^{+0.16}_{-0.13} \pm 0.02$ ~ps,

 $\tau_H = 2.07 ^{+0.58}_{-0.46} \pm 0.03$ ~ps,

 $\Delta \Gamma /\overline \Gamma   = 0.65 ^{+0.25}_{-0.33} \pm 0.01$.\\

Figure \ref{fig:cdf_dg} shows  the scan of the likelihood function
for $\Delta \Gamma /\overline \Gamma$.
Pseudoexperiments tossed with $\Delta \Gamma /\overline \Gamma =0$
yield the betting odds for observing the above results at
1/315. For $\Delta \Gamma /\overline \Gamma = 0.12$ (SM prediction,
which has recently been updated to 0.14$\pm$0.05~\cite{dg_un}) the betting odds are
1/84.

\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0.3\textheight,width=8.2cm]  {cdf_scan-dg-un.eps}

\vspace*{-1cm}
\caption{Scan of the likelihood function
for $\Delta \Gamma /\overline \Gamma$ (CDF).
}
\label{fig:cdf_dg}
\end{figure}

D\O\ has used a novel technique to  measure the lifetime ratio
of the charged and neutral $B$ mesons, exploiting the large
semileptonic sample. $B$ hadrons were reconstructed in the channels
$B\rightarrow \mu^+ \nu D^*(2010)^-X$, which are dominated by $B^0$ decays,
and  $B\rightarrow \mu^+ \nu D^0X$, which are dominated by $B^+$ decays.
The lifetime ratio was
obtained from the variation of the ratio of the number of events in these two
processes at different decay lengths.
The result is \\

$\tau(B^+)/\tau(B^0_d)$   =  1.093$\pm$0.021$\pm$0.022.      ~(D\O)

\subsection{Towards $B_s$ mixing}

Measurement of the $B_s$ oscillation frequency via ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing
will provide an important constraint on the CKM matrix. The oscillation
frequency is proportional to the mass difference between the mass eigenstates,
$\Delta m_s$, and is related to the CKM matrix through
$\Delta m_s \propto |V_{tb}V_{ts}|$. When combined with the
$B_d$ mass difference, $\Delta m_d$ it helps in extraction of $|V_{td}|$,
and thereby the CP violating phase.

As a benchmark for future $B_s$ oscillation measurement, both groups
study  $B_d$ mixing, gaining an understanding of the different components
of a $B$ mixing analysis (sample composition, flavor tagging, vertexing,
asymmetry fitting). For a sample of partially reconstructed decays
$B\rightarrow D^*(2010)^+\mu^-X$, D\O\ obtains
$\Delta m_d = 0.506 \pm 0.055 (stat) \pm  0.049 (syst))$ ps$^{-1}$ and
$\Delta m_d = 0.488 \pm 0.066 (stat) \pm  0.044 (syst))$ ps$^{-1}$
when employing  opposite side muon tagging and the same side tagging,
respectively.

The CDF result for semileptonic channels is
$\Delta m_d = 0.536 \pm 0.037 (stat) \pm  0.009 (s.c.) \pm 0.015 (syst)$ ps$^{-1}$.
CDF also reports a result on $B$ oscillations using fully reconstructed
decays:
$\Delta m_d = 0.526 \pm 0.056 (stat) \pm  0.005 (syst))$ ps$^{-1}$.

Reconstructing $B_s$ decays into different final states is another
important
 step in the ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing analysis.
Thanks to the  large muon and tracking coverage,   D\O\ is accumulating
a  high statistics sample of semileptonic $B_s$ decays.
D\O\ reconstructs the $B_s \rightarrow D^+_s \mu^- X$ decays, with
$D^+_s \rightarrow \phi \pi^+ $ and
$D^+_s \rightarrow K^* K^+ $,
at a rate of $\approx$ 40(25) events per pb$^{-1}$,  respectively.
Figure \ref{fig:d0_bsdsphipi} shows the mass distribution of the
$D^+_s \rightarrow \phi \pi$ candidates.

\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=0.3\textheight,width=8.0cm]  {blds-250.eps}
\vspace*{-1.2cm}
\caption{  $D^+_s \rightarrow \phi \pi^+$  signal. (D\O)}
\label{fig:d0_bsdsphipi}
\end{figure}

\begin{figure}[htb]
\vspace*{-10mm}
\hspace*{-4mm}
\includegraphics[height=0.35\textheight,width=7.9cm]  {cdf_Bs-DsPi-PhiPi.eps}

\vspace*{-1.0cm}
\caption{ $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$  signal.  (CDF)}
\label{fig:cdf_bsdsphipi}
\end{figure}

CDF has clean signals for fully hadronic, flavor-specific  $B_s$ decays,
providing the best sensitivity to $B_s$ oscillations at high
$\Delta m_s$. Figure \ref{fig:cdf_bsdsphipi} shows the signal for
the best channel, $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$.

\clearpage

\subsection{Rare decays}

The purely leptonic decays $B_{d,s}^0 \rightarrow \mu^+
\mu^-$ are flavor-changing neutral current (FCNC) processes.
In the standard model, these decays are forbidden at the tree level and
proceed at a very low rate through higher-order diagrams.
The latest SM prediction~\cite{sm_ref3}
is ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-)=(3.42\pm 0.54)\times
10^{-9}$, where the error is dominated by non-perturbative uncertainties. The
leptonic branching fraction of the $B_d^0$ decay is suppressed by CKM matrix elements $|V_{td}/V_{ts}|^2$
leading to a predicted SM branching fraction of $(1.00\pm0.14)\times 10^{-10}$.
The best published experimental bound (Fig.~\ref{fig:cdf_bsmumu})
 for the branching fraction
of $B^0_s$ $(B^0_d)$ is presently
${\cal B}(B^0_s \, (B^0_d) \rightarrow \mu^+\mu^-)<7.5\times 10^{-7}\,
(1.9\times 10^{-7})$ at the 95\% C.L.~\cite{cdfII}.
The decay amplitude of $B^0_{d,s} \rightarrow \mu^+ \mu^-$ can be
significantly enhanced in some extensions of the SM.

\begin{figure}[htb]
\includegraphics[height=8.3cm,width=7.9cm]  {cdfbsmumu_results_prl.eps}

\vspace*{-1cm}
\caption{Invariant mass for the events passing all requirements. (CDF)}
\label{fig:cdf_bsmumu}
\end{figure}

Assuming no contributions
from the decay $B^0_d\rightarrow \mu^+\mu^-$ in the signal region,
D\O\  finds the conservative upper limit on the branching fraction
to be ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-) \leq 4.6\times 10^{-7}$
at the 95\% C.L. (Fig.~\ref{fig:d0_bsmumu}).

\begin{figure}[htb]
\includegraphics[height=5.0cm,width=8.0cm]  {B06F03.eps}
\vspace*{-1cm}
\caption{Invariant mass for the events  passing all requirements. (D\O)}
\label{fig:d0_bsmumu}
\end{figure}


\section{Introduction}

Ever since the discovery of cosmic rays \citep{hess1912},
the origin and the acceleration mechanism
up to more than $\sim$TeV
have been long-standing problems.
\citet{koyama1995} discovered synchrotron X-rays
from shock fronts of a supernova remnant (SNR), SN~1006,
and suggested that
shocks of SNRs are cosmic ray accelerators.
Several SNRs have recently been categorized
into synchrotron X-ray emitters
(G347.3$-$0.5: \citet{koyama1997,slane1999},
RCW~86: \citet*{bamba2000,borkowski2001b},
and G266.6$-$1.2: \citet{slane2001}).
Recently,
a significant number of SNRs with synchrotron X-rays have been discovered
by the {\it ASCA} Galactic plane survey
\citep{bamba2001,ueno2003,bamba2003a,yamaguchi2004}.
These discoveries provide good evidence for the cosmic ray acceleration
at the shocked shell of SNRs.
The most plausible process of the cosmic ray acceleration is
the diffusive shock acceleration (DSA)
\citep[e.g.][]{bell1978,blandford1978,drury1983,%
blandford1987,jones1991,malkov2001},
which can accelerate particles on the shock
into a power-law distribution,
similar to the observed spectrum of cosmic rays
showering on the earth.

Apart from the global success of DSA,
there are still many remaining problems.
We have not yet fully understood detailed but important information,
such as the maximum energy of particles,
the configuration of magnetic fields,
the injection efficiency from thermal plasma to accelerated particles,
and the acceleration and de-acceleration history of particles
around the shock fronts, and so on.
This partly comes from the fact that
there is no information
about the spatial distribution of accelerated electrons of practical objects,
which strongly reflects the above uncertainties.

\citet{bamba2003b} considered
the spatial distribution of emission from accelerated particles
as new information
in order to understand the acceleration on the shock in SNRs;
they observed the synchrotron X-ray emitting shell of SN~1006
with the excellent spatial resolution of the {\it Chandra} X-ray observatory
\citep[see also][]{long2003},
and found that
the emitting regions of synchrotron X-rays are
incredibly thin ("filaments"),
about 0.01--0.1~pc on the upstream side
and 0.06--0.4~pc on the downstream side,
at 2.18~kpc distance \citep{winkler2003}.
This result constrains the DSA theory and magnetic-field configurations;
both an efficient acceleration scenario with strong magnetic field
parallel to the shock normal
and an inefficient one with a weak and perpendicular magnetic field
are allowed \citep{yamazaki2004b,berezhko2003}.

In the present work,
we searched for nonthermal filaments in other historical SNRs
(Cas~A, Kepler's remnant, Tycho's remnant, and RCW~86)
and measured their scale widths and spectral parameters.
It was also investigated
whether or not the age of the SNR is correlated with
the scale width and/or spectral parameters of filaments.
If it is, such correlations might describe the time evolution
of these parameters in a single SNR.
Although the characteristics of the SNR should be considered,
those correlations may also  become  clues to understand
the time evolution of the maximum energy of electrons,
the magnetic field configuration,
the injection efficiency, and so on.
This paper is organized as follows.
We summarize the observation details of five SNRs with {\it Chandra}
in \S\ref{sec:obs}.
\S\ref{sec:individual} briefly describes
the results of the observations of the four SNRs.
In \S\ref{sec:discussion}
we discuss the origin of emission (\S\ref{sec:origin}),
our analysis of individual SNRs
(\S\ref{sec:individual2}),
the time evolution of the spatial and spectral parameters of filaments
together with previous analysis of SN~1006,
which have already been analyzed in \citet{bamba2003b}
(\S\ref{sec:evo}).
We also briefly deal with the time evolution of energy densities of
magnetic field and cosmic rays in this section.
A summary of our results is given in \S\ref{sec:sum}.

\section{Observations}
\label{sec:obs}

We used the {\it Chandra} archival data of the ACIS
of five historical SNRs
(Cas~A, Kepler, Tycho, SN~1006, and RCW~86),
as listed in Table~\ref{tab:obs_log}.
Note that this data set includes almost historical shell-type SNRs.
The satellite and the instrument are described
by \citet{weisskopf2002} and \citet{garmire2000}, respectively.
Data acquisition from the ACIS was made in the Timed-Exposure Faint mode.
The data reductions and analysis were made using the {\it Chandra}
Interactive Analysis of Observations (CIAO) software version 2.3.
Using the Level 2 processed events provided by the pipeline
processing at the {\it Chandra} X-ray Center,
we selected {\it ASCA} grades 0, 2, 3, 4, and 6, as the X-ray events.
The effective exposure for each observation is also given
in Table~\ref{tab:obs_log}.

\section{Individual Results}
\label{sec:individual}

\subsection{Cas~A (G111.7$-$2.1)}

Although the remnant's precise age is still uncertain,
the most probable record of the explosion
is SN~1680, made by \citet{flamsteed1725},
which we adopt in this paper.
The distance to the remnant was estimated to be $3.4_{-0.1}^{+0.3}$~kpc
by \citet{reed1995}
using the measurement of optical proper motions.
The average radio index of the whole remnant at 1~GHz is
$0.78$ \citep{green2004}.

Cas~A emits hard X-rays up to greater than 10~keV,
which were detected by {\it RXTE} \citep{allen1997},
{\it Beppo}SAX \citep{favata1997},
and OSSE onboard {\it GRO} \citep{the1996}.
\citet{vink2003} has already found and analyzed
one of the nonthermal filamentary structures
with {\it Chandra}.
There are many other filaments thinner than that which they analyzed.
These filaments are considered to be profiles of a thin sheet-like structure,
as suggested by \citet{hester1987};
then, the scale width of these filaments is
the upper-limit width of the real thickness
in the radial direction of the sheets.
Therefore, we consider other filaments in this paper,
which are located much nearer to the aim-point (less than 25 arcsec)
and are much thinner than those analyzed by \citet{vink2003}.

The upper-left panel of Figure~\ref{fig:images}
shows the south-eastern side of Cas~A in the 5.0--10.0~keV band,
in which there are a lot of filamentary structures.
These filaments are more clumpy than those in SN~1006 \citep{bamba2003b}.
We selected two filaments as shown in the panel,
which are straight enough to be made profiles
and near the aim-point.

Figures~\ref{fig:profiles}(a) and (b) show
the profiles of two filaments.
In order to estimate the scale width of these filaments,
we used a simple, empirical fitting model
for the apparent profiles,
which is the same
as used in the previous analysis in the SN~1006 case by \citet{bamba2003b}:
\begin{equation}
f(x) = \left\{
        \begin{array}{rlr}
        A\exp \left(-|\frac{x_0-x}{w_u}|\right) & {\rm on\ the\ upstream\ side} \\
        A\exp \left(-|\frac{x_0-x}{w_d}|\right) & {\rm on\ the\ downstream\ side},
        \end{array}
\right.\label{model}
\end{equation}
where $A$ and $x_0$ are the flux and position at the emission peak,
respectively.
The quantities $w_u$ and $w_d$ represent the e-folding widths on
the upstream and the downstream sides of the apparent emission peak,
respectively
(hereinafter, "$u$" and "$d$" represent
upstream and downstream sides, respectively).
We treat the best-fit values smaller than 0.8~arcsec as the upper limit,
because they highly suffer the influence of point spread function (PSF),
which is about 0.5 arcsec
within $\sim$4~arcmin radius from the aim point.
The fittings were made with profiles in the 5.0--10.0~keV,
and were accepted statistically
with the best-fit models and parameters
shown in Figures~\ref{fig:profiles}(a) and (b), and Table~\ref{tab:fila_image}.

We then made the spectra of the filaments within the
scale widths
($x_0-w_u \leq x \leq x_0+w_d$)
in Figures~\ref{fig:profiles}(a) and (b).
The background spectra were made from
the just off-filament downstream sides with the same width of the filaments.
We compared the Si line photon counts
with those in the continuum (5.0--10.0~keV) band
in these regions,
and found that photons in this band include
thermal photons in a ratio of only less than 25\%,
under the assumption that
all photons in the background regions are thermal,
and both regions have a thermal component
with the same physical parameters, except for their flux.
Therefore, we treated all the 5.0--10.0~keV photons
in our analysis of Cas~A,
as a part of the hard X-rays
reported in previous observations \citep{vink2003}.

The background-subtracted spectrum accumulated from the two filaments
is shown in the upper-left panel of Figure~\ref{fig:spectra},
which is featureless with no line-like structure,
and extends to the hard X-ray side,
which was fitted with both an absorbed thin thermal plasma model
in non-equilibrium ionization (NEI)
calculated by \citet{borkowski2001a}
("{\tt NEI}" model),
and an absorbed power-law model.
The absorption column was subsequently calculated using the cross sections by
\citet{morrison1983} with the solar abundances \citep{anders1989}.
Both models were well fitted statistically with best-fit curves and parameters,
shown in Figure~\ref{fig:spectra} and Table~\ref{tab:fila_spec}.
We also applied the {\tt SRCUT} model,
which represents the synchrotron emission from electrons of power-law
distribution with exponential roll-off in a homogeneous magnetic field
\citep{reynolds1999}.
The spectral index at 1~GHz was fixed to be $0.78$
according to a report by \citet{green2004}.
This model also well reproduces the spectra
with best-fit values shown in Table~\ref{tab:fila_spec}.
The spectra are much harder than those of the filament analyzed by
\citet{vink2003}.

\subsection{Kepler's Remnant (G4.5+6.8)}

Kepler's remnant is the second youngest SNR in our Galaxy
recorded by human beings,
which appeared in 1604 \citep{kepler1606}.
\citet{delaney2002} observed this SNR with VLA,
and found the spatial variation of spectral index from $0.85$ to $0.6$
with a mean value of $0.7$.
The distance to the remnant is $4.8\pm1.4$~kpc
using the H~I observations with VLA \citep{reynoso1999}.
\citet*{petre1999} found hard X-rays (above 10~keV) from Kepler
with {\it RXTE},
and {\it XMM-Newton} pointed out that
at least the south-eastern edge emits nonthermal X-rays
\citep{cassamchenai2004}.

The upper-right panel of Figure~\ref{fig:images} shows
a 4.0--10.0~keV band {\it Chandra} image of the eastern rim of Kepler.
We can see very sharp structures on the outer edge of the remnant,
especially bright on the eastern side,
which is similar to the filaments in SN~1006.
In order to carry out spatial and spectral analysis of these structures,
we selected two filaments, as shown in the panel,
which are $<$160~arcsec distant from the aim-point.
Figures~\ref{fig:profiles} (c) and (d) show
the profiles of the filaments
in the 4.0--10.0~keV band.
They have sharp edges both on the upstream and downstream sides of
the emission peak,
and eq.(\ref{model}) well represents their profiles
with best-fit curves and parameters
summarized in the panels and Table~\ref{tab:fila_image}.

We obtained the spectra of these filaments
in the same way of the Cas~A case.
Background regions were selected just
from the downstream sides of the filaments,
which are free from any other structures.
The intensities of iron and silicon lines and continuum photons
in the 4.0--10.0~keV band were compared,
and it was found that
more than 90\% of the photons in the 4.0--10.0~keV band
are the hard X-ray component
reported in previous observations \citep{petre1999,cassamchenai2004},
with the same assumptions as in the Cas~A case.
Thus the derived scale widths represent those of the hard X-ray emission.
The background-subtracted spectrum accumulated from these filaments
is shown in the upper-right panel of Figure~\ref{fig:spectra},
which shows a hard and line-less feature.
We fitted each spectrum
with {\tt NEI} and power-law models
with an absorption.
Both models are well fitted to the data
with the best-fit models and parameters
shown in the panel and Table~\ref{tab:fila_spec}.
The {\tt SRCUT} model was also applied
with the spectral index at 1~GHz fixed to be $0.70$,
derived by \citet{delaney2002} as the average value on the eastern side.
This model also well represents the data
with the best-fit parameters listed in Table~\ref{tab:fila_spec}.

\subsection{Tycho's Remnant (G120.1+1.4)}

Tycho is a remnant of a supernova
that exploded in 1572 \citep{tycho1573}.
The distance to the remnant is estimated to be 1.5--3.1~kpc,
from the proper motion of the optical filaments \citep{kamper1978}
and the shock velocity estimated by \citet{ghavamian2001}.
We adopt the most popular value, 2.3~kpc.
\citet{katzstone2000} investigated
the variations of the radio spectral indices,
and found that the emission in the outer rim
shows a trend that brighter clumps have a flatter spectral index
than the average index of $s = 0.52\pm 0.02$,
maybe due to either SNR blast waves and ambient medium interactions or
internal inhomogeneities of the magnetic field within the remnant.

Hard X-ray emission has been unmistakably detected from Tycho,
up to 30~keV,
with {\it HEAO 1} \citep{pravdo1979},
{\it Ginga} \citep{fink1994},
and {\it RXTE} \citep{petre1999}.
{\it Chandra} and {\it XMM-Newton} also observed Tycho's remnant,
and the good spatial resolutions of instruments onboard these satellites
show the spatial variety of X-ray emission
\citep{hwang2002,decourchelle2001}.
\citet{hwang2002} claimed that
there is a thin structure encircling the SNR,
the spectrum of which has lines with low equivalent widths.

In order to be clear as to
whether the thin structure is a hard X-ray emitter or not,
spatial and spectral analysis were conducted in the same way
as in the cases for Cas~A and Kepler.
The lower-left panel of Figure~\ref{fig:images} shows
the north-western part of the remnant in the 2.0--10.0~keV band.
The thin filament-like structure suggested by \citet{hwang2002}
can be seen.
Figure~\ref{fig:profiles} (e)--(i) shows
profiles of the filaments selected in the panel,
which are located within 120~arcsec from the aim-point.
Very sharp rises and decays can be seen in all of the filaments.
The fittings with eq.(\ref{model}) were performed only
around the outermost peak of hard X-ray emission,
because thermal photons dominate in the downstream regions
(the right side of each panel),
and were accepted statistically
with the best-fit models and parameters shown
in these panels and Table~\ref{tab:fila_image}.

Their spectra were obtained in the same way as in the Cas~A case.
Background spectra were accumulated from just the downstream side
of the filaments.
The lower-left panel of Figure~\ref{fig:spectra} shows
the background-subtracted spectra,
which were combined with those of filaments in each CCD chip.
Since both are hard and have no line-like structure,
we fitted them with {\tt NEI}, power-law, and {\tt SRCUT} models
with an absorption.
We fixed the spectral index at 1~GHz to be $0.52$,
following \citet{katzstone2000}.
All three models were accepted statistically
with similar probability,
with the best-fit values listed in Table~\ref{tab:fila_spec}.
We compared the Si and S lines and the 2.0--10.0~keV continuum band intensity
in the source spectrum with those in background region,
and it was found that
only less than 6\% of the thermal photons contribute in the 2.0--10.0~keV band.
Therefore,
it is likely that
the all photons in the 2.0--10.0~keV band can be treated as pure nonthermal.

\subsection{RCW~86 (G315.4$-$2.3)}

RCW~86 was identified as
being the oldest historical SNR by \citet{clark1977}.
They suggested that
a progenitor of the remnant was reported in AD~185
in Later Han dynasty records \citep{fanyou432},
although \citet{chin1994} claimed some doubt
about the record.
The spectral index at 1~GHz and the distance was measured
to be $0.6$ by \citet{caswell1975} and 2.8~kpc \citep{rosado1996},
respectively.

Strong hard X-rays have been detected from the remnant
by {\it Ginga} \citep{kaastra1992} and {\it RXTE} \citep{petre1999}.
Using {\it ASCA} data,
\citet{bamba2000} and \citet{borkowski2001b} independently found that
the hard X-rays show a nonthermal feature,
and concentrate on the south-western (SW) shell of the SNR.
The {\it Chandra} observation with excellent spatial resolution
reveals that
the regions emitting nonthermal X-rays are very clumpy
and different from those emitting thermal X-rays \citep{rho2002}.

The lower-right panel of Figure~\ref{fig:images} represents
a close-up view of the SW shell in the 2.0--10.0~keV band.
Clumpy structures can be seen,
which have been already indicated as nonthermal filaments by \citet{rho2002}.
We selected two filaments for our analysis,
which are bright in the hard band image
and are located $<$120~arcsec from the aim-point.
Their profiles from the region indicated in the panel were made,
as shown in Figures~\ref{fig:profiles}(j) and (k).
They show a sharp rise and a rather slow decay in the upstream and
the downstream sides of the emission peak,
respectively.
The fitting with eq.(\ref{model}) was carried out in the 2.0--10.0~keV band,
and was accepted statistically with best-fit models and the values
shown in the panels and Table~\ref{tab:fila_image}.

The spectra were made in the same way as in the former cases.
The background regions were selected from the downstream side of the filaments
without other structures.
The emission is free from thermal photons,
according to previous studies,
especially in such a hard band \citep{rho2002}.
The lower-right panel of Figure~\ref{fig:spectra} shows
the background-subtracted spectrum,
which was accumulated from both filaments.
The spectral fittings were made with the three models
({\tt NEI}, power-law, and {\tt SRCUT}) with an absorption.
We adopted the spectral index at 1~GHz
with the value  $0.6$ derived by \citet{caswell1975}.
All of the models well reproduced the data
with the best-fit profiles and values given
in the panel and Table~\ref{tab:fila_spec}.

\section{Discussion}
\label{sec:discussion}

\subsection{Emission Origin of Filaments}
\label{sec:origin}

Table~\ref{tab:fila_spec} summarizes the spectral fittings
for the filaments in the four SNRs
to thermal ({\tt NEI}) and nonthermal (power-law and {\tt SRCUT}) models.
Because all models give a similar reduced $\chi^2$,
we cannot distinguish which model is the best statistically
to reproduce the spectra of the filaments.
The thermal model fittings for all cases, however, require
an unusually high temperature and a low abundance
for young (ejecta dominated) SNRs.
On the other hand,
the best-fit photon indices in the power-law model are $\sim$2--3,
which are similar to hard X-rays in other SNRs
\citep[e.g.,][]{koyama1995}.
It reminds us of the synchrotron emission from electrons
with a power-law distribution of index of $\Gamma=1.5$ or softer,
suggesting that these filaments are acceleration sites of electrons
via the DSA mechanism.
The fitting of the {\tt SRCUT} model requires
a roll-off frequency of about 1~keV,
as shown in Table~\ref{tab:fila_spec}.
In order to check the influence of uncertainty of the radio index $s$,
we fitted the spectra with the frozen $s$ of 0.5 and 0.9.
For the $s=0.5$ cases,
which is the theoretical minimum indicated by the DSA,
$\nu_{rolloff}$ remains unchanged within a factor of two.
On the other hand, the fittings with $s=0.9$,
which implies that the de-acceleration is very efficient,
require the change of $\nu_{rolloff}$ in the order of 1.
However, the best-fit normalization in the radio band becomes
larger than 10\% of the total flux from the whole SNR,
which is unrealistic for such small regions.
Moreover, in the Kepler case, the radio map shows that
the value of $s$ of the rim is smaller than
that of the inner region \citep{delaney2002},
although it changes from 0.85 to 0.6.
This result suggests that the outer rims of SNRs
are acceleration sites and de-acceleration may not occur yet,
in other words, $s$ is around 0.6.
For the Tycho case,
\citet{katzstone2000} indicated that
the spectral index of filaments in outer rim is around 0.5,
indicating that $\nu_{rolloff}$ derived in our analysis for the SNR
is roughly correct.
These results indicates that
the X-ray photons shown in Figure~\ref{fig:images} are emitted by
electrons with the highest energy in each SNR.

It can be argued that the thermal spectra in source and background
regions may be the same.
The line emission from the NEI plasma changes
as the ionization time scale ($n_et$) varies.
We take the background regions just behind the source regions,
and their widths are much small (1--5\% of SNR radii).
The electron density $n_e$ may not change so drastically
in such thin downstream regions.
Since we take the background regions just behind the source regions,
and their widths are much small (1--5\% of SNR radii),
the electron density $n_e$ does not so drastically change
in the downstream regions.
Thus, the difference of $n_et$ between the background and source regions
may not be large
enough to clarify the difference between their thermal spectra.

All of the filaments have much smaller scale widths
than that derived from the Sedov self-similar solution ($=R/12$),
on both the upstream and downstream sides,
which is similar to the previous results in the SN~1006 case
\citep{bamba2003b}.
We should consider the influence of PSF,
especially for the filaments in Cas~A and Tycho,
where $w_u$ are as small as the PSF (see Table~\ref{tab:fila_image}).
The PSF enlarges the apparent $w_u$ and $w_d$.
Thus, our result may be the upper limit of the physical scale width.
Together with the fact that
our sample covers almost historical SNRs with an X-ray shell,
it may be suggested that
all relative young ($\la$2000~years) SNRs accelerate electrons
in very thin regions around shock fronts.

\subsection{Comments on Individual SNRs}
\label{sec:individual2}

\subsubsection{Cas~A}

We analyzed filaments thinner than that by \citet{vink2003},
and the former has harder spectra than the latter.
This fact may imply that
the magnetic field around our filaments is stronger than that around
the filament by \citet{vink2003}.
Based on analysis of the filament-like structure in Cas~A by
\citet{vink2003},
\citet{berezhko2004}
estimated the magnetic field around the structure
to be $\sim 500~\mu$G,
generated by the cosmic rays themselves \citep{lucek2000}.
Our thinner filaments may have
a larger magnetic field according to their discussion.

\subsubsection{Kepler's Remnant}

As for Kepler, since there is no report about nonthermal filaments,
this is the first analysis of them.
\citet{cassamchenai2004} reported that
south-eastern edge of the SNR emits nonthermal X-rays
with {\it XMM-Newton}.
Their spectral parameters are consistent with ours.
Therefore, the region by \citet{cassamchenai2004}
may be a part of our filaments.

\subsubsection{Tycho's Remnant}

Our spectral results are roughly consistent with
the results by \citet{hwang2002},
although the regions for spectral analysis are slightly different,
except for the S abundance in the {\tt NEI} model.
The difference may come
mainly from selection of the different background regions;
we chose them not from out of the SNR, but from just at the downstream side
of the filaments.
We could thus extract pure nonthermal emission.

\subsubsection{RCW~86}

We re-confirmed the result of spectral analysis by \citet{rho2002}.
A similar discussion about the magnetic field in the SN~1006 case by
\citet{yamazaki2004b} may be possible,
although the filaments
are very clumpy and we may have to consider the curvature effect
suggested by \citet{berezhko2003}.
Together with the shock velocity of 562~km~s$^{-1}$ \citep{ghavamian2001}
and results of spatial and spectral analysis (see previous sections),
we estimated the magnetic field around the filaments is
$\sim$~4--12~$\mu$G \citep[see also][]{bamba2004}.
\citet{watanabe2003} reported on TeV gamma-ray observations
of the SW region of this remnant using the CANGAROO telescope.
According to our results,
inverse Compton TeV gamma-rays
may be detected
with the current TeV $\gamma$-ray telescopes.
The estimated flux of inverse Compton emission is
$\sim6\times10^{-13}(B_{\rm d}/10\ \mu{\rm G})^{-2}$~ergs~s$^{-1}$cm$^{-2}$.

\subsection{Correlations between the age and spatial and spectral parameters}
\label{sec:evo}

We found thin and nonthermal filaments in four historical SNRs,
which may be acceleration sites of high-energy electrons.
Their scale widths and spectral parameters include a lot of information
about physical parameters inevitable
for understanding acceleration on the shock,
such as the gyro radius of electrons, the magnetic field configuration,
and so on,
as already suggested by \citet{bamba2003b} in the SN~1006 case.
Furthermore, a correlation between the age of SNRs
and the spatial and spectral parameters
might represent their time evolution in a single SNR,
although we should consider the characteristics of the individual source.
In this section,
we consider some correlation between the age and the physical parameters
as a rough discussion about the time evolution.

Figure~\ref{fig:evolution}(a) represents
the average scale width
weighted by the flux of each filament
 ($w_u$ and $w_d$ in the unit of pc)
as a function of the age ($t_{age}$ in the unit of year)
for five historical SNRs;
we analyzed four and one was from the previous results of SN~1006
\citep{bamba2003b}.
The distances to the SNRs are fixed to previously estimated values
(see \S\ref{sec:individual}).
The panel shows that in all SNRs,
both $w_u$ and $w_d$ are quite smaller than the SNR radii.
Moreover, there is a positive correlation between $w_d$ and $t_{age}$;
the scale width becomes larger as the SNR becomes older.
We fitted them with a power-law function
as a tentative model,
and obtain the relations such as:
\begin{equation}
w_d = 3.0^{+0.7}_{-0.6}\times 10^{-6} t_{age}{}^{1.60_{-0.30}^{+0.01}}\ \ ,
\label{eq:wd}
\end{equation}
with the reduced $\chi^2$ of 9.68/3.
Hereinafter, the errors indicate 90\% confidence regions.
The index is larger than that for the shock width of thermal gas
derived from the Sedov solution (=2/5),
although we reached no definite conclusion
because the fittings were rejected statistically.

Figures~\ref{fig:evolution}(b) and (c) show the plots of
photon indices in the power-law fittings vs. the age ($t_{age}$)(b)
and $\nu_{rolloff}$ (Hz) of the {\tt SRCUT} model vs. $t_{age}$ (c),
respectively.
The spectrum seems to grow softer
in the very young phase (less than 500~years),
whereas the spectral parameters remain constant at 1000--2000~years old.
Since the error is too large, especially in the young phase,
we gave up modeling the time evolution of the spectral parameters.

We searched for another parameter that has a clear correlation with time,
and found a tentative function, such as
\begin{equation}
{\cal B} = \nu_{rolloff}w_d{}^{-2}\ \ .
\end{equation}
Figure~\ref{fig:evolution}(d) represents
the relation between $t_{age}$ and ${\cal B}$.
It shows monotonous decay,
and a power-law fitting (${\cal B}=Ct_{age}{}^\alpha$)
was accepted statistically
with best-fit values of
\begin{eqnarray}
C &=& 2.6_{-1.4}^{+1.2}\times 10^{27}\ \, \\
\alpha &=& -2.96_{-0.06}^{+0.11}\ \ , \label{eq:f_index}
\end{eqnarray}
with a reduced $\chi^2$ of 2.03/3, respectively.
This result implies that
the function ${\cal B}$ might include some physical quantities
that evolve with time.
We checked the uncertainty of ${\cal B}$ due to the change of $s$
as already discussed in \S\ref{sec:origin},
and found that
the best-fit value does not change
although error regions become 10 times larger.
Therefore, the uncertainty of $s$ cannot change our results.

Here, we make a scenario in order to explain the time evolution of ${\cal B}$.
Our assumption is that the spatial profile found in our present analysis
reflects that of accelerated electrons \citep{bamba2003b,yamazaki2004b}.
The maximum energy of accelerated electrons, $E_{max}$, is determined by
the age of the SNR, or the synchrotron energy loss,
such as $t_{\rm acc}\sim{\rm min}\{t_{\rm age},t_{\rm loss}\}$,
where $t_{\rm acc}$ and $t_{\rm loss}$
are the acceleration and the synchrotron loss time scale,
respectively \citep{yamazaki2004b}.
Then, we simply obtain the scale width of
the accelerated electrons in the radial direction, $\Delta_d$, as
$\Delta_d\sim v_d \times{\rm min}\{t_{\rm age},t_{\rm loss}\}
\sim v_d t_{\rm acc}$,
where $v_d$ is the downstream fluid velocity.
The acceleration time $t_{\rm acc}$ is on the order of $K/v_s^2$,
where $v_s$ and $K$
are the shock velocity and
the diffusion coefficient, respectively.
The quantity $K$ is assumed to
be proportional to the gyro radius of the accelerated electrons,
$r_g=E_{max}/eB_d$, where $B_d$ is
the downstream magnetic field  \citep{drury1983,yamazaki2004b}.
As a result, we find
$\Delta_d\sim K/v_s\propto B_d{}^{-1}E_{max}v_s{}^{-1}$.
Because of the projection (curvature) effect,
observed apparent scale width in the downstream region, $w_d$,
differs from $\Delta_d$ \citep[e.g.,][]{berezhko2004}.
The ratio $w_d/\Delta_d$ depends on uncertainty of
radial profile of accelerated electron distribution downstream.
It is found that for sufficiently small
$\Delta_d$ compared with the local curvature radius,
the ratio $w_d/\Delta_d$ is about 7, 5, or 1
when we assume the exponential, Gaussian, or the top-hat form of the
downstream radial profile, respectively.
Hence it may be assumed that the ratio $w_d/\Delta_d(\sim3)$ remains
unchanged within a factor of 3, so we find
$w_d\propto\Delta_d\propto B_d{}^{-1}E_{max}v_s{}^{-1}$.
The value of $\nu_{rolloff}$ is proportional to $B_dE_{max}{}^2$
\citep{reynolds1999}.
Therefore, the quantity ${\cal B}$ is proportional to $B_d{}^3v_s{}^2$.
The time evolution of ${\cal B}$ may represent
that of $B_d{}^3v_s{}^2$.

Let us consider the time evolution of the magnetic field and the shock velocity
as $B_d \propto t_{age}{}^{-\ell}$ and $v_s \propto t_{age}{}^{-m}$.
The index, $m$, changes from 0 to $3/5$
as SNRs evolve from the free-expansion phase to the Sedov phase.
Together with the observational fact of eq.(\ref{eq:f_index}),
the index $\ell$ takes the value of 0.57--1.02.
Our result suggests that
the magnetic field decreases as the SNR ages.
This is the first observational implication
of the time evolution of the magnetic field in the SNR.

The most interesting case is $\ell=0.6$,
which is allowed when SNRs are in the Sedov phase ($m=0.6$).
In this case, since the density of the thermal plasma is almost constant,
the energy density of the magnetic field ($u_B \propto B_d{}^2$),
and the kinetic and thermal energy densities of the shock ($u_{th} \propto$
shock temperature
and $u_{kin} \propto v_s{}^2$),
evolves according to the same time dependency (index = $-1.2$).
The time evolution of the energy density of accelerated protons ($u_{p}$)
may also be discussed
from the relations by \citet{lucek2000} as
\begin{eqnarray}
u_{p} &=& \frac{v_s}{v_A} u_B \propto B_dv_s
\propto t_{age}{}^{-1.2}\ \ ,
\end{eqnarray}
where $v_A \equiv B/\sqrt{4\pi\rho}$ and $\rho$ are the Alfv{\'e}n velocity
and the fluid density,
suggesting that
accelerated protons in an SNR evolve with the same energy density evolutions
as that of the other energy carriers.
Since \citet{bamba2003b} indicated that
these energy densities are roughly in equipartition with each other,
our result implies that
they evolve while maintaining equipartition.
\citet{bell2001} suggested theoretically that
the magnetic field on the shock evolves
according to the same time dependency as
the shock velocity,
which is consistent with our result.

There remain many unsolved problems that we should consider.
The value of $w_d$ depends on many parameters that we have ignored,
such as the curvature effect,
the angle between the magnetic field and the shock normal,
the degree of turbulence, and so on.
Solving these problems is beyond our work,
which is going to be considered in \citet{yamazaki2004a}.

\section{Summary}
\label{sec:sum}

We have conducted systematic spectral and spatial analysis of
filamentary structures
in historical SNRs for the first time.
A summary of our results is as follows:

\begin{enumerate}
\item
We discovered very thin filaments on the outer edges of four historical SNRs:
Cas~A, Kepler, Tycho, and RCW~86.
The scale widths were measured for the first time;
they are below or the same as the PSF size of {\it Chandra} and 0.01--0.4~pc
on the upstream and downstream sides of the emission peak, respectively.
\item
Although
both thermal and nonthermal spectral models can reproduce
the spectra of the filaments,
the former is unlikely since
it requires an unrealistic high temperature and low abundances.
The photon indices ($\sim$2--3) and $\nu_{rolloff}$
($\sim 10^{16}$--$10^{18}$~Hz) of nonthermal models suggest
the synchrotron X-ray emission from electrons accelerated via DSA,
in a similar way to the SN~1006 case.
\item
The value of $w_d$ (see eq.(\ref{eq:wd})) becomes
larger as the SNR ages.
\item
The spectra of filaments might become softer as the SNR ages
in the phase of $t_{age} \la 500$~years,
whereas it does not change when $1000 \la t_{age} \la 2000$~years,
although there is no definite conclusion because of a lack of statistics.
\item
The time evolution of a tentative function
${\cal B} = \nu_{rolloff}/w_d{}^2$
was analyzed,
and it has a clear negative correlation
with $t_{age}$ with the index of $\sim -3$.
This may imply that
the downstream magnetic field decreases
with the index ($\ell$) of 0.6--1.0.
When $\ell=0.6$,
energy densities of magnetic field, shock (thermal and kinetic),
and cosmic rays evolve
while keeping equipartition with each other.
\end{enumerate}

\acknowledgements

Our particular thanks are due to
M. Hoshino, S. Inutsuka, K. Makishima, and F. Takahara,
for their fruitful discussions and comments.
We also thank an anonymous referee
for helpful comments.
R.Y. is supported by JSPS Research Fellowship for Young Scientists.