Anosov Diffeomorphisms on a Product of Surfaces

TL;DR

This paper proves that certain complex dynamical systems called transitive Anosov diffeomorphisms with a product structure cannot exist on the product of two high-genus surfaces, especially when homotopic to a product of pseudo-Anosov maps.

Contribution

It establishes a non-existence result for transitive Anosov diffeomorphisms with a specific structure on products of surfaces with genus ≥ 2.

Findings

No transitive Anosov diffeomorphism with the specified structure exists on the product of two high-genus surfaces.

The result applies to diffeomorphisms homotopic to a product of pseudo-Anosov maps.

The proof constrains the types of dynamical systems possible on complex surface products.

Abstract

We show that there is no transitive Anosov diffeomorphism with the global product structure, which is homotopic to a product of pseudo-Anosov diffeomorphisms, on a product of two closed surfaces each of which has genus greater than or equal to two.