On cusps and flat tops

TL;DR

This paper introduces non-invertible Pesin theory for cusp maps with flat critical points, expanding understanding of invariant measures and dynamics in systems with unbounded derivatives and flat critical points.

Contribution

It develops a new Pesin theory framework applicable to cusp maps, including those with flat critical points, generalizing previous results and analyzing invariant measures.

Findings

Maps with flat critical points may lack absolutely continuous invariant measures.

The theory applies to $C^{1+ ext{epsilon}}$ maps with unbounded derivatives.

Generalizes Benedicks and Misiurewicz's results to broader classes of maps.

Abstract

We develop non-invertible Pesin theory for a new class of maps called cusp maps. These maps may have unbounded derivative, but nevertheless verify a property analogous to $C^{1+\epsilon}$. We do not require the critical points to verify a non-flatness condition, so the results are applicable to $C^{1+\epsilon}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.