119135787
Bifurcation currents and equidistribution on parameter space
In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.
116973610
Notions of Convexity
The first two chapters of the book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets and sets which are convex for supports or singular supports with respect to a differential operator. In addition the convexity conditions which are relevant for local or global existence of holomorphic solutions of holomorphic differential equations are discussed, leading up to Trepreau's theorem on sufficiency of condition (psi) for microlocal solvability in the analytic category.
126049801
Dynamics of automorphisms of compact complex surfaces
Recent results concerning the dynamics of holomorphic diffeomorphisms of compact complex surfaces are described, that require a nice interplay between algebraic geometry, complex analysis, and dynamical systems. RÉSUMÉ. Nous décrivons quelques résultats récents concernant la dynamique des difféomorphismes holomorphes des surfaces complexes compactes. Ceci nécessite des outils de géométrie algébrique, d’analyse complexe et de systèmes dynamiques.
121503131
Entropy properties of rational endomorphisms of the Riemann sphere
In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.
2288224
Dynamics of rational maps: a current on the bifurcation locus
Let fλ : P 1 → P1 be a family of rational maps of degree d > 1, parametrized holomorphically by λ in a complex manifold X. We show that there exists a canonical closed, positive (1,1)-current T on X supported exactly on the bifurcation locus B(f) ⊂ X. If X is a Stein manifold, then the stable regime X − B(f) is also Stein. In particular, each stable component in the space Polyd (or Ratd) of all polynomials (or rational maps) of degree d is a domain of holomorphy.
118147879
Bifurcation currents in holomorphic dynamics on
Abstract We use pluri-potential theory to study the bifurcations of holomorphic families {ƒ λ}λ∈X of rational maps on or endomorphisms of . To this purpose we establish some formulas for L(ƒ λ) and ddcL(ƒ λ) where L(ƒ λ) is the sum of the Lyapunov exponents of ƒ λ with respect to the maximal entropy measure. We show that the bifurcation current ddcL(ƒ λ) both detects the instability of repulsive cycles and the interaction between critical and Julia sets. For families of rational maps of degree d, we introduce a bifurcation measure defined by (ddcL(ƒ λ))2d−2 add study its first properties. In particular, we show that the support of this measure is contained in the closure of the set of rational maps having 2d − 2 distinct Cremer-Cycles. This approach yields to a purely potential-theoretic proof of Mañé-Sad-Sullivan theorem and, moreover, allows us to extend it.
11020188
CUBIC POLYNOMIALS: A MEASURABLE VIEW OF PARAMETER SPACE
We study the ne geometric structure of bifurcation currents in the parameter space of cubic polynomials viewed as dynamical systems. In particular we prove that these currents have some laminar structure in a large region of parameter space, reecting the possibility of quasiconformal deformations. On the other hand, there is a natural bifurcation measure, supported on the closure of rigid parameters. We prove a strong non laminarity statement relative to this measure.
15535084
Distribution of polynomials with cycles of a given multiplier
Abstract In the space of degree d polynomials, the hypersurfaces defined by the existence of a cycle of period n and multiplier eiθ are known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.
15085659
Random walks, Kleinian groups, and bifurcation currents
Let (ρλ)λ∈Λ be a holomorphic family of representations of a finitely generated group G into PSL(2,ℂ), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λ describing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on ℙ1. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μ on G.We show that under natural assumptions on μ, the support of the bifurcation current coincides with the bifurcation locus of the family. We also prove that the bifurcation current describes the asymptotic distribution of several codimension 1 phenomena in parameter space, like accidental parabolics or new relations, or accidental collisions between fixed points.
15695154
Dynamique des applications d'allure polynomiale
We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem: 
 
Let f:U→V be a proper holomorphic map from an open set U⋐V onto a Stein manifold V. Assume f is of topological degree dt⩾2. Then there is a probability measure μ supported on K:=⋂n⩾0f−n(V) satisfying the following properties: 
 
(1) The measure μ is invariant, K-mixing, of maximal entropy logdt. 
 
(2) If J is the Jacobian of f with respect to a volume form Ω then ∫logJdμ⩾logdt. 
 
(3) For every probability measure ν on V with no mass on pluripolar sets dt−n(fn)∗ν⇀μ. 
 
(4) If the p.s.h. functions on V are μ-integrables (μ is PLB), then 
 
(a) The Lyapounov exponents for μ are strictly positive; 
 
(b) μ is exponentially mixing; 
 
(c) There is a proper analytic subset E0 of V such that f−1(E0)⊂E0 and for z∉E, μzn:=dt−n(fn)∗δz⇀μ where E=⋃n⩾0fn(E0); 
 
(d) The measure μ is a limit of Dirac masses on the repelling periodic points. 
 
The condition μ is PLB is stable under small pertubation of f. This gives large families where it is satisfied.
14560744
Lyapunov exponents and bifurcation current for polynomial-like maps
We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.
17137812
Remarks on Iterated Cubic Maps
This articl e discusses the dynamics of iterated cubic maps from the real or complex line to itself and will describe the geography of the parameter space for such maps. It is a rough survey with few precise statements or proofs and depends strongly on work by Douady, Hubbard, Branner and Rees.
18618158
On Rational Maps with Two Critical Points
This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points.
122884
Cubic Polynomial Maps with Periodic Critical Orbit, Part I
The parameter space for cubic polynomial maps has complex dimension 2. Its non-hyperbolic subset is a complicated fractal locus which is difficult to visualize or study. One helpful way of exploring this space is by means of complex 1-dimensional slices. This note will pursue such an exploration by studying maps belonging to the complex curve Sp consisting of all cubic maps with a superattracting orbit of period p . Here p can be any positive integer. A preliminary draft of this paper, based on conversations with Branner, Douady and Hubbard, was circulated in 1991 but not published. The present version tries to stay close to the original; however, there has been a great deal of progress in the intervening years. (See especially (Faught 92), (Branner and Hubbard 92), (Branner 93), (Roesch 99, 06), and (Kiwi 06).) In particular, a number of conjectures in the original have since been proved; and new ideas have made sharper statements possible. We begin with the period 1 case. Section 2 studies the dynamics of a cubic polynomial map F which has a superattracting fixed point, and whose Julia set J(F ) is connected. The filled Julia set of any such map consists of a central Fatou component bounded bounded by a Jordan curve, together with various limbs sprouting off at internal angles which are explicitly described.
117793839
Complex Analytic Sets
The theory of complex analytic sets is part of the modern geometric theory of functions of several complex variables. Traditionally, the presentation of the foundations of the theory of analytic sets is introduced in the algebraic language of ideals in Noetherian rings as, for example, in the books of Herve [23] or Gunning-Rossi [19]. However, the modern methods of this theory, the principal directions and applications, are basically related to geometry and analysis (without regard to the traditional direction which is essentially related to algebraic geometry). Thus, at the beginning of this survey, the geometric construction of the local theory of analytic sets is presented. Its foundations are worked out in detail in the book of Gunning and Rossi [19] via the notion of analytic cover which together with analytic theorems on the removal of singularities leads to the minimum of algebraic apparatus necessary in order to get the theory started.
119147714
Fatou directions along the Julia set for endomorphisms of CP^k
Not much is known about the dynamics outside the support of the maximal entropy measure $\mu$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(\mu)$. The Julia set is the support of the so-called Green current $T$, so it admits a natural filtration by the supports of the exterior powers of $T$. For $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ there are at least $(k-q)$ "Fatou directions" in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions.
73674301
On the dynamics of rational maps
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1983, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
7105480
Canonical heights, transfinite diameters, and polynomial dynamics
Abstract Let ϕ(z ) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating ϕ gives rise to a dynamical system and a corresponding canonical height function ĥϕ , as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of ϕ over various completions of K, and we apply this formula to give a generalization of Bilu’s equidistribution theorem for sequences of points whose canonical heights tend to zero.
6177780
Distribution of rational maps with a preperiodic critical point
Let $\{f_\lambda\}$ be a family of rational maps of a fixed degree, with a marked critical point $c(\lambda)$. Under a natural assumption, we first prove that the hypersurfaces of parameters for which $c(\lambda)$ is periodic converge as a sequence of positive closed $(1,1)$ currents to the bifurcation current attached to $c$ and defined by DeMarco. We then turn our attention to the parameter space of polynomials of a fixed degree $d$. By intersecting the $d-1$ currents attached to each critical point of a polynomial, Bassaneli and Berteloot obtained a positive measure $\mu_{\rm bif}$ of finite mass which is supported on the connectedness locus. They showed that its support is included in the closure of the set of parameters admitting $d-1$ neutral cycles. We show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials (i.e., of Misiurewicz points).
122599838
Families of Rational Maps and Iterative Root-Finding Algorithms
In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms. In the context of conformal dynamics, our main result is the following: a stable algebraic family of rational maps is either trivial (all its members are conjugate by Mobius transformations), or affine (its members are obtained as quotients of iterated addition on a family of complex tori). Our classification of generally convergent algorithms follows easily from this result. As another consequence of rigidity, we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices. This implies that bounded analytic functions nearly separate points on the moduli space of a rational map.
16096199
Polynomial Di eomorphisms of C 2 . IV : The Measure of Maximal Entropy and Laminar
The simplest holomorphic dynamical systems which display interesting behavior are the polynomial maps of C. The dynamical study of these maps began with Fatou and Julia in the 1920's and is currently a very active area of research. If we are interested in studying invertible holomorphic dynamical systems, then the simplest examples with interesting behavior are probably the polynomial diieomorphisms of C 2. These are maps f : C 2 ! C 2 such that the coordinate functions of f and f ?1 are holomorphic polynomials. For polynomial maps of C the algebraic degree of the polynomial is a useful dynamical invariant. In particular the only dynamically interesting maps are those with degree d greater than one. For polynomial diieomorphisms we can deene the algebraic degree to be the maximum of the degrees of the coordinate functions. This is not, however, a conjugacy invariant. Friedland and Milnor FM] gave an alternative deenition of a positive integer deg f which is more natural from a dynamical point of view. If deg f > 1, then deg f coincides with the minimal algebraic degree of a diieomorphism in the conjugacy class of f. As in the case of polynomial maps of C, the polynomial diieomorphisms f with deg(f) = 1 are rather uninteresting. We will make the standing assumption that deg(f) > 1. For a polynomial map of C the point at innnity is an attractor. Thus the \recurrent" dynamics can take place only on the set K consisting of bounded orbits. A normal families argument shows that there is no expansion on the interior of K so \chaotic" dynamics can occur only on J = @K. This set is called the Julia set and plays a major role in the study of polynomial maps. For diieomorphisms of C 2 each of the objects K and J has three analogs. Corresponding to the set K in one dimension, we have the sets K + (resp. K ?) consisting of the points whose orbits are bounded in forward (resp. backward) time and the set K := K + \ K ? consisting of points with bounded total orbits. Each of these sets is invariant and K is compact. As is in the one dimensional case, recurrence can occur only on the set K. Corresponding to the set J in dimension one, we have the sets J := @K , and the set J …
54186648
Solution of the Fatou - Julia problem on wandering domains
Disclosed is an apparatus for machining the bolster bowl surface, the bolster pocket surfaces, the fulcrum pin hole surface and the bolster gib surfaces of railroad truck bolsters. The apparatus includes a base, a fixture mounted for rotation on the base and clamps on the fixture for clamping the bolster rigidly to the fixture. A locating template is utilized to properly position the bolster on the fixture, and once properly positioned and clamped, the fixture can be rotated and locked in a plurality of different positions to permit machining of the various surfaces. To machine the bolster pocket surfaces, the fulcrum pin hole surface, and the bolster gibs, a first machining (milling) head is provided that is movable along three perpendicular axes. To accomplish machining of the bolster bowl surface, a second machining (boring/facing) head is provided that is pre-aligned so that once the bolster is properly positioned with the locating template, the second machining head need only be moved along one axis to machine the bolster bowl. The second machining head has a combination boring and facing tool mounted thereon which can accomplish both a boring and facing operation simultaneously to machine the bolster bowl in a single operation.
13913132
Lyapunov Exponents for Surface Group Representations
Let $${(\rho_\lambda)_{\lambda \in \Lambda}}$$(ρλ)λ∈Λ be a holomorphic family of representations of a surface group $${\pi_1(S)}$$π1(S) into $${\mathrm{PSL}(2, \mathbb{C})}$$PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space Λ, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the ddc of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincaré metric. We show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in Λ. For instance, the random hypersurfaces of Λ defined by the condition that a random closed geodesic on S is mapped under ρλ to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current. The proofs are based on our previous work (Deroin and Dujardin, Invent Math 190:57–118, 2012), and on a careful control of a discretization procedure of the Brownian motion.
118842460
Hyperbolic Manifolds and Discrete Groups
Preface.-Three-dimensional Topology.-Thurston Norm.-Geometry of the Hyperbolic Space.-Kleinian Groups.-Teichmuller Theory of Riemann Surfaces.-Introduction to the Orbifold Theory.-Complex Projective Structures.-Sociology of Kleinian Groups.-Ultralimits of Metric Spaces.-Introduction to Group Actions on Trees.-Laminations, Foliations and Trees.-Rips Theory.-Brooks' Theorem and Circle Packings.-Pleated Surfaces and Ends of Hyperbolic Manifolds.-Outline of the Proof of the Hyperbolization Theorem.-Reduction to The Bounded Image Theorem.-The Bounded Image Theorem.- Hyperbolization of Fibrations.-The Orbifold Trick.-Beyond the Hyperbolization Theorem References.-Index.
17082039
Cusps are dense
We show cusps are dense in Bers' boundary for Teichmuller space. The proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic Riemann surface.
14653811
Distribution of periodic points of polynomial diffeomorphisms of C2
This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of $\C^2$: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$.
17474419
Puiseux series polynomial dynamics and iteration of complex cubic polynomials
Nous considerons la completion L du corps des series formelles de Puiseux et nous etudions les polynomes a coefficients dans L en tant que systemes dynamiques. Nous donnons une description complete de l'espace dynamique et l'espace des parametres des polynomes cubiques a coefficients dans L. Nous demontrons que la dynamique cubique sur L et sur C sont intimement liees. Plus precisement, nous montrons que certains elements de L correspondent de maniere naturelle a des series de Fourier de fonctions analytiques presque periodiques (au sens de Bohr) qui parametrisent (a l'infini) les classes quasi-conformes des polynomes complexes cubiques non renormalisables. Nos techniques s'appuient sur des idees introduites par Branner et Hubbard pour l'etude des polynomes cubiques complexes.
119639585
Puiseux series dynamics of quadratic rational maps
We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.