Thick hyperbolic repelling invariant Cantor set and wild attractor

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.

16 pages, 5 figures

References in corpus (1)

• α-limit sets and Lyapunov function for maps with one topological attractor