Kupka-Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

TL;DR

The paper constructs explicit examples of non-uniformly hyperbolic diffeomorphisms at the boundary of uniform hyperbolicity, featuring cubic heteroclinic tangencies and Kupka-Smale properties, with implications for symbolic dynamics and equilibrium states.

Contribution

It provides a novel explicit construction of non-uniform hyperbolic diffeomorphisms with heteroclinic tangencies and Kupka-Smale properties at the boundary of hyperbolicity.

Findings

Existence of non-uniform hyperbolic diffeomorphisms with cubic heteroclinic tangencies.

Diffeomorphisms are Kupka-Smale for a broad set of choices.

Map is conjugate to a subshift of finite type, ensuring unique equilibrium states.

Abstract

We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a certain degree of freedom for the choice of the second one. For a non-countable set of choices, this leaf is not periodic and the diffeomorphism is Kupka-Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. As a consequence of our construction, the map is H\"older conjugated to a subshift of finite type, thus every H\"older potential admits a unique associated equilibrium state.