Cases where


$sup\_{\mu \in E(T)} h\_\mu(T)
\neq
\sup\_{\mu \in M(T)} h\_\mu(T)$.


Background
----------


For a topological space $X$,
let $T: X \to X$ be a continuous application.
Then, call the set of $T$-invariant probability measures
$M(T)$, and call the set of $T$-ergodic (probability) measures
$E(T)$.
It is evident that $E(T) \subset M(T)$.
But it may happen that $M(T) = \emptyset$.
For example, take $X = \mathbb{R}$ and $T(x) = x+1$.


Since an ergodic measure is invariant, it is immediate that


$
 \begin{equation\*}
 \sup\_{\mu \in E(T)} h\_\mu(T)
 \leq
 \sup\_{\mu \in M(T)} h\_\mu(T).
 \end{equation\*}
$


The question is whether equality holds or not.
When $X$ is compact, it is well known that equality holds.
In this case, it is a consequence of Jacobs' Theorem, which
states that for any $\mu \in M(T)$, there exists a measure
$\tau$, over the set $E(T)$, such that


$
 \begin{equation\*}
 h\_\mu(T)
 =
 \int\_{E(T)} h\_m(T) d\tau(m).
 \end{equation\*}
$


When $X$ is compact (locally compact, in fact),
the above equation is a consequence of
[Choquet Representation Theorem](http://en.wikipedia.org/wiki/Choquet_theory)
and the [Krein-Milman Theorem](http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem).
(See, for example, Theorem 8.4 from
Walters, P. *An Introduction to Ergodic Theory*)


Now, when $X$ is not necessarily compact, but it is a Borel subset of
a compact metrizable set $\widetilde{X}$, Pesin and Pitskel'
argue in their
*Topological Pressure and the Variational Principle for Noncompact Sets*,
at the end of page 310:
(I will rename the spaces and applications in order
to conform to this post's notation.)



> 
> We may assume that measure $\mu$ is ergodic.
>  In fact, consider the partition $\eta$ of $X$ into
>  ergodic components $X\_s,\, s \in S$, of measure $\mu$.
>  Denote by $\mu\_s$ the measures on $X\_s$
>  (then $T \* \mu\_s = \mu\_s$), and by $\nu$ the measure on the
>  quotient space $X / \eta$.
>  Then $h\_\mu(T) = \int\_{Y/\eta} h\_m(T) d\nu(m)$.
> 
> 
> 


As far as I understand, $Y/\eta$ is just the same as $E(T)$,
since each ergodic component is associated with an ergodic measure.
And for the same reason, $\nu$ is just our $\tau$.
So, what is being stated is the validity of


$
 \begin{equation\*}
 h\_\mu(T)
 =
 \int\_{E(T)} h\_m(T) d\tau(m),
 \end{equation\*}
$


which in turns implies the equality


$
 \begin{equation\*}
 \sup\_{\mu \in M(T)} h\_\mu(T)
 =
 \sup\_{\mu \in E(T)} h\_\mu(T).
 \end{equation\*}
$


In Pitskel' and Pesin's paper, $T$ is not even supposed to be the restriction
to $X$ of a continuous transformation
$\widetilde{T}: \widetilde{X} \to \widetilde{X}$.


Questions
---------


1. How do I prove that when $X$ is a Borel subset of a compact
metrizable space $\widetilde{X}$ and $T$ is a continuous application
$T: X \to X$, then for any $\mu \in M(T)$, there exists
a measure $\tau$ over $E(T)$ such that
$h\_\mu(T) = \int\_{E(T)} h\_m(T) d\tau(m)$?
2. In case the answer to question "1" is negative,
is there a prove for the specific case where $T$ is the restriction
of a continuous application
$\widetilde{T}: \widetilde{X} \to \widetilde{X}$?
3. Do you know *nice* examples of transformations
of measurable spaces where $E(T) = \emptyset$ while
$M(T) \neq \emptyset$?




---


PS: This is my first post to MathOverflow.
This is really exciting! :-)



Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.
You make a pretty common mistake here - your question has nothing to do with continuity, compactness etc. and belongs entirely to the measure category. What matters here is that you have a measure preserving transformation of a probability Lebesgue space (sometimes these spaces are also called standard probability spaces - there is a nice wikipedia article about them) and a measurable partition of this space pointwise invariant with respect to the transformation (particular, and, essentially, the only interesting case: the partition into ergodic components of the transformation). Then the entropy of the transformation is the integral of the entropies of its restrictions to the elements of the partition (the formula which appears in Pesin and Pitskel'). In fact, it should not be attributed to them - a much better reference (actually, for the whole entropy theory as well) is Rokhlin's lecture notes on entropy theory in Russian Math Surveys 1967 (the formula we discuss is in section 8.11 there). It takes care of your questions (1) and (2). As for (3), any invariant probability measure has a unique (mod 0) decomposition into ergodic invariant measures (once again, this is an entirely measure category property), so that there are **no** (nice or not so nice) examples of transformations $T$ with $E(T)=\emptyset$ while $M(T)\neq\emptyset$.


