Smooth Anosov Katok Diffeomorphisms With Generic Measure

TL;DR

This paper constructs numerous Anosov-Katok diffeomorphisms exhibiting non-ergodic measures, mixing properties, and diverse topological features, including explicit sets of generic points with interesting Hausdorff dimensions.

Contribution

It introduces new examples of Anosov-Katok diffeomorphisms with complex measure-theoretic and topological properties, expanding understanding of their dynamical behavior.

Findings

Existence of non-ergodic generic measures

Construction of systems with various mixing properties

Explicit sets of generic points with notable Hausdorff dimensions

Abstract

We construct a plethora of Anosov-Katok diffeomorphisms with non-ergodic generic measures and various other mixing and topological properties. We also construct an explicit collection of the set containing the generic points of the system with interesting values of its Hausdorff dimension.