Exceptional sets for nonuniformly expanding maps

DOI10.1088/0951-7715/29/4/1238zbMATH Open1355.37068arXiv1506.00676OpenAlexW3104248376MaRDI QIDQ2805234

Sara Campos, Katrin Gelfert

Publication date: 10 May 2016

Published in: Nonlinearity (Search for Journal in Brave)

Abstract: Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dynamical dimension and the Hausdorff dimension coincide. We discuss also the case of a general conformal $C^{1+alpha}$ dynamical system and, in particular, certain multimodal interval maps on their Julia set.

Full work available at URL: https://arxiv.org/abs/1506.00676

zbMATH Keywords

Hausdorff dimensiontopological entropyexceptional sets

Mathematics Subject Classification ID

Conformal densities and Hausdorff dimension for holomorphic dynamical systems (37F35) Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets (37F10) Topological entropy (37B40) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25) Dynamical systems involving maps of the interval (37E05) Dimension theory of smooth dynamical systems (37C45)

Related Items (4)

Exceptional sets for nonuniformly hyperbolic diffeomorphisms ⋮ Exceptional sets for geodesic flows of noncompact manifolds ⋮ Lyapunov spectrum for exceptional rational maps ⋮ Exceptional sets for average conformal dynamical systems

This page was built for publication: Exceptional sets for nonuniformly expanding maps

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q2805234)