Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity

Abstract

In this paper, we consider a one-parameter family $F_{\lambda }$ ofcontinuous maps on $\mathbb{R}^{m}$ or $\mathbb{R}^{m}\times \mathbb{R}^{k}$with the singular map $F_{0}$ having one of the forms (i) $F_{0}(x)=f(x),$(ii) $F_{0}(x,y)=(f(x),g(x))$, where $g:\mathbb{R}^{m}\rightarrow \mathbb{R}^{k}$ is continuous, and (iii) $F_{0}(x,y)=(f(x),g(x,y))$, where $g:\mathbb{R}^{m}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}$ is continuous andlocally trapping along the second variable $y$. We show that if $f:\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ is a $C^{1}$ diffeomorphism having atopologically crossing homoclinic point, then $F_{\lambda }$ has positivetopological entropy for all $\lambda $ close enough to $0$.