Repellers for real analytic maps
TLDR
In this article, it was shown that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f.Abstract:
The purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see corollary 5).read more
Citations
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Complex analytic dynamics on the Riemann sphere
TL;DR: In this paper, the authors define a dynamique dynamique de Fatou et Julia, which is defined as the dynamique of points periodiques repulseurs of a group of polynomes.
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Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors
TL;DR: In this paper, the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to non-hyperbolic systems, and the generalized partial dimensions of the invariant measure and the distribution of effective Liapunov exponents for hyperbolic attractors are investigated.
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Area distortion of quasiconformal mappings
TL;DR: In this paper, it has been shown that K-quasiconformal mappings are locally H51der continuous with exponent 1/K. The function is defined as follows:
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The thermodynamic formalism for expanding maps
TL;DR: In this article, the spectral properties of transfer operators and corresponding analytic properties of the generating function are discussed, and new results are proved and some natural conjectures are proposed, and applications to Julia sets are also discussed.
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Related aspects of positivity in Riemannian geometry
TL;DR: In this article, a variete de Riemann on etudie le lien entre le point λ 0 du L 2 -spectre le plus proche de zero for l'extension de Friedrich d'un operateur semi-defini symetrique Δ and le taux exponentiel de transition of a operateur de Markoff conservant la positivite.
References
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BookDOI
Equilibrium states and the ergodic theory of Anosov diffeomorphisms
TL;DR: Gibbs Measures and Gibbs measures have been used in this article to define Axiom a Diffeomorphisms for general Thermodynamic Formalism and Ergodic Theory of Axiom-a-Diffeomorphism.
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Hausdorff dimension of quasi-circles
TL;DR: In this paper, the authors present a legal opinion on the applicability of commercial and impression systématiques in the context of the Copyright Agreement of the Publications Mathématique de l'I.H.É.S.
Book
Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics
TL;DR: In this paper, the authors present a book aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, and the study of differentiable dynamical systems, notably Anosov diffeomorphisms and flows.
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Zeta-functions for expanding maps and Anosov flows
TL;DR: In this article, a meromorphic zeta function for Anosov flows is shown to be meromorphic if the flow and its stable-unstable foliations are real-analytic.