Hyperbolic Homoclinic Classes of Surface Endomorphism
Matéo Ghezal (USP)

In the last few years, symbolic codings of non-uniformly hyperbolic systems have been developed extensively and have proved very fruitful, considerably improving our understanding of equilibrium states in this context. Such symbolic techniques are particularly effective when restricted to specific invariant subsets of the dynamics, namely homoclinic classes, which consist of points exhibiting transverse intersections between their stable and unstable manifolds. Indeed, any homoclinic class can be coded by a transitive Markov shift, and many remarkable properties of equilibrium states can be derived from this coding. It therefore seems very natural today to study homoclinic classes in non-uniformly hyperbolic settings. Buzzi, Crovisier, and Sarig have provided a thorough understanding of these sets in the case of smooth surface diffeomorphisms. A natural next step is to understand homoclinic classes in the setting of surface local diffeomorphisms. Here, homoclinic classes can exhibit two very different behaviours, hyperbolic or expanding, depending on the unstable dimension. We will see that the set of hyperbolic homoclinic classes with sufficiently large entropy is finite. We will also discuss partial results on hyperbolic homoclinic classes of positive entropy, where the two behaviours, expanding and hyperbolic, can interact.