Some sufficient conditions for transitivity of Anosov diffeomorphisms

TL;DR

This paper establishes that certain Jacobian conditions imply transitivity in $C^2$-Anosov diffeomorphisms, simplifying the proof of classical theorems and showing that regularity of points guarantees transitivity.

Contribution

It proves that Jacobian conditions imply transitivity in $C^2$-Anosov diffeomorphisms, removing the need for transitivity as an initial assumption in related theorems.

Findings

Jacobian condition $Jf^n(p)=1$ implies transitivity.

Regular points condition ensures transitivity.

Simplifies classical theorems in Sinai-Ruelle-Bowen theory.

Abstract

Given a $C^2$- Anosov diffemorphism $f: M \rightarrow M,$ we prove that the jacobian condition $Jf^n(p) = 1,$ for every point $p$ such that $f^n(p) = p,$ implies transitivity. As application in the celebrated theory of Sinai-Ruelle-Bowen, this result allows us to state a classical theorem of Livsic-Sinai without directly assuming transitivity as a general hypothesis. A special consequence of our result is that every $C^2$-Anosov diffeomorphism, for which every point is regular, is indeed transitive.