Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles

TL;DR

This paper demonstrates that blenders, which enable robust dynamical phenomena, naturally occur near heterodimensional cycles in smooth diffeomorphisms and establishes a generic dichotomy in local dynamics based on arithmetic properties.

Contribution

It proves the natural existence of blenders near heterodimensional cycles and characterizes the local dynamics dichotomy using arithmetic properties of conjugacy moduli.

Findings

Blenders exist without perturbation near heterodimensional cycles.

A generic dichotomy in local dynamics is established: either infinite blenders or no other orbits.

The existence of blenders depends on arithmetic properties of conjugacy moduli.

Abstract

Blenders are special hyperbolic sets used to produce various robust dynamical phenomena which appear fragile at first glance. We prove for $C^r$ diffeomorphisms ($r=2,\dots,\infty,\omega$) that blenders naturally exist (without perturbation) near non-degenerate heterodimensional cycles of coindex-1, and the existence is determined by arithmetic properties of moduli of topological conjugacy for diffeomorphisms with heterodimensional cycles. In particular, we obtain a $C^r$-generic dichotomy for dynamics in any small neighborhood $U$ of a non-degenerate heterodimensional cycle: either there exist infinitely many blenders accumulating on the cycle, forming robust heterodimensional dynamics in most cases, or there are no orbits other than those constituting the cycle lying entirely in $U$.