Fast growth of the number of periodic points arising from heterodimensional connections

TL;DR

This paper demonstrates that under certain conditions, small perturbations of diffeomorphisms with heterodimensional cycles can create an abundance of periodic points, leading to super-exponential growth in their number.

Contribution

It establishes conditions on non-linearities and Schwarzian derivatives that ensure rapid proliferation of periodic points near heterodimensional cycles.

Findings

Super-exponential growth of periodic points in generic diffeomorphisms.

Construction of perturbations creating flat periodic points.

Examples confirming the necessity of the conditions.

Abstract

We consider C^r-diffeomorphisms of a compact smooth manifold having a pair of robust heterodimensional cycles where r is a positive integer or infinity. We prove that if certain conditions about the signatures of non-linearities and Schwarzian derivatives of the transition maps are satisfied, then by giving C^r arbitrarily small perturbation, we can produce a periodic point at which the first return map in the center direction is C^r-flat. As a consequence, we will prove that C^r-generic diffeomorphisms in the neighborhood of the initial diffeomorphism exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions on non-linearities and the Schwarzian derivatives.