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Almost blenders and parablenders

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory Topological dynamics

Published online by Cambridge University Press:  24 March 2022

SÉBASTIEN BIEBLER
SÉBASTIEN BIEBLER*
Affiliation:
Université de Paris, IMJ-PRG, 8 Place Aurélie Nemours, Paris 75205, France
*
e-mail: sebastien.biebler@imj-prg.fr
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Abstract

A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds robustly contains an open set. Introduced by Bonatti and Díaz in the 1990s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper we introduce analogous notions in a measurable setting. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of $\mathbb {R}^2$ leaving invariant the continuation of a hyperbolic basic set. When an inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger, and complements previous works on the construction of blenders by Avila, Crovisier, and Wilkinson or by Moreira and Silva. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak, and Urbański, we study families of skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.

Keywords

blenderparablenderprevalence

MSC classification

Primary: 37C05: Smooth mappings and diffeomorphisms 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37B40: Topological entropy 37C45: Dimension theory of dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 4 , April 2023 , pp. 1087 - 1128
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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