Thick hyperbolic repelling invariant Cantor set and wild attractor

TL;DR

This paper constructs a smooth unimodal map with a thick hyperbolic repelling invariant Cantor set containing a wild attractor, showing such complex dynamics are dense in a certain parameter space.

Contribution

It introduces a new method to create $C^1$ unimodal maps with wild attractors using Denjoy-like surgery, expanding understanding of complex invariant sets.

Findings

The set $D$ of parameters is dense in (1, 2].

Constructed maps have thick hyperbolic repelling Cantor sets with wild attractors.

Such maps cannot be $C^{1+eta}$ due to measure properties of the invariant set.

Abstract

Let $D$ be the set of $\beta \in (1, 2]$ such that $f_\beta$ is a symmetric tent map with finite critical orbit. For $\beta \in D$, by operating Denjoy like surgery on $f_{\beta}$, we constructed a $C^1$ unimodal map $\tilde{g}_\beta$ admitting a thick hyperbolic repelling invariant Cantor set which contains a wild Cantor attractor. The smoothness of $\tilde{g}_\beta$ is ensured by the effective estimation of the preimages of the critical point as well as the prescribed lengths of the inserted intervals. Furthermore, $D$ is dense in $(1, 2]$, and $\tilde{g}_\beta$ can not be $C^{1+\alpha}$ because the hyperbolic repelling invariant Cantor set of $C^{1+\alpha}$ map has Lebesgue measure equal to zero.