Transitivity of real Anosov diffeomorphisms

TL;DR

This paper proves the transitivity of real Anosov diffeomorphisms, demonstrating that under certain conditions, these systems exhibit a form of topological mixing, with implications for their dynamical behavior.

Contribution

The paper establishes transitivity for real Anosov diffeomorphisms by analyzing conformal hyperbolic distances and holonomy of stable and unstable curves, extending understanding of their dynamical structure.

Findings

Proved transitivity of real Anosov diffeomorphisms.

Established existence of a conformal hyperbolic distance with well-defined lengths.

Connected stable/unstable curve properties to global holonomy.

Abstract

We prove the transitivity of real Anosov diffeomorphisms, which are Anosov diffeomorphisms where stable and unstable spaces decompose into a continuous sum of invariant one-dimensional sub-spaces with uniform contraction/expansion over the ambient manifold. We prove that if a stable/unstable curve has a well-defined length in a conformal hyperbolic distance, then it has a globally defined holonomy. We exhibit a conformal hyperbolic distance with well-defined length of stable/unstable curves for each real Anosov diffeomorphism.