On codimension one partially hyperbolic diffeomorphisms

TL;DR

This paper proves that codimension one partially hyperbolic diffeomorphisms on tori are derived from linear Anosov systems, are uniquely integrable, and exhibit intrinsic ergodicity, confirming Katok's conjecture in this setting.

Contribution

It establishes the structure, integrability, and ergodic properties of codimension one partially hyperbolic diffeomorphisms on tori, confirming a conjecture about ergodic measures.

Findings

Support on f4nf4s diffeomorphisms

Local unique integrability

Intrinsic ergodicity and validation of Katok's conjecture

Abstract

We show that every codimension one partially hyperbolic diffeomorphism must support on $\mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is intrinsically ergodic, and the A. Katok's conjecture about the existence of ergodic measures with intermediate entropies holds for it.