On non-uniformly hyperbolicity assumptions in one-dimensional dynamics

TL;DR

This paper reformulates the backward contracting property in one-dimensional dynamics, linking it to orbit expansion of critical values for certain complex polynomials and smooth interval maps, enhancing understanding of hyperbolic behavior.

Contribution

It provides an equivalent formulation of the backward contracting property based on orbit expansion, applicable to specific classes of complex polynomials and smooth interval maps.

Findings

Equivalent formulation of backward contracting property

Applicable to finitely renormalizable complex polynomials

Includes all smooth interval maps with non-flat critical points

Abstract

We give an essentially equivalent formulation of the backward contracting property, defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values, for complex polynomials of degree at least two which are at most finitely renormalizable and have only hyperbolic periodic points, as well as for all smooth interval maps with non-flat critical points.