Jianrong Xiao

Thick hyperbolic repelling invariant Cantor set and wild attractor

Let $D$ be the set of $β\in (1, 2]$ such that $f_β$ is a symmetric tent map with finite critical orbit. For $β\in D$, by operating Denjoy like surgery on $f_β$, we constructed a $C^1$ unimodal map $\tilde{g}_β$ admitting a thick hyperbolic repelling invariant Cantor set which contains a wild Cantor attractor. The smoothness of $\tilde{g}_β$ 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 $\tilde{g}_β$ can not be $C^{1+α}$ because the hyperbolic repelling invariant Cantor set of $C^{1+α}$ map has Lebesgue measure equal to zero.