Almost blenders and parablenders

TL;DR

This paper introduces the concepts of almost blenders and parablenders, extending the classical notions to a measurable setting, and demonstrates their existence under certain entropy and contraction conditions, partially confirming Berger's conjecture.

Contribution

It generalizes blenders and parablenders to a measurable context, providing new conditions for their existence in dynamical systems.

Findings

Almost blenders have positive Lebesgue measure local unstable sets.

Conditions involving entropy and contraction imply the existence of almost blenders.

The results partially confirm Berger's conjecture on robust dynamics.

Abstract

A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and D{\'i}az in the 90s, 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 point of view. 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 R2 leaving invariant the continuation of a hyperbolic basic set. When some 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. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak and Urba{\'n}ski, we study families of fiberwise unipotent skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.