Structural Stability for Fibrewise Anosov Diffeomorphisms on Principal Torus Bundles

TL;DR

This paper proves that fibre-preserving hyperbolic diffeomorphisms on principal torus bundles are topologically conjugate to linear maps along the fibres, revealing a form of structural stability.

Contribution

It establishes a topological conjugacy for fibrewise Anosov diffeomorphisms on principal torus bundles, extending understanding of their stability and linearization.

Findings

Fibre-preserving hyperbolic diffeomorphisms are conjugate to linear maps.

The conjugacy preserves the fibre structure.

The result applies to compact principal torus bundles.

Abstract

We show a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.