# Foliations and Conjugacy II: The Mendes Conjecture for Time-One Maps of Flows

## TL;DR

This paper proves Mendes' conjecture for the specific case where the Anosov diffeomorphism is the time-one map of a flow, showing that only linear automorphisms and translations are topologically conjugate in this setting.

## Contribution

It establishes the validity of Mendes' conjecture for time-one maps of flows using a new theorem on invariant foliations.

## Key findings

- Mendes' conjecture holds for time-one maps of flows.
- Invariant foliations under time-one maps are characterized.
- Only linear automorphisms and translations are conjugate in this case.

## Abstract

A diffeomorphism $f:\mathbb{R}^2\to\mathbb{R}^2$ in the plane is Anosov if it has a hyperbolic splitting at every point of the plane. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations (the existence of Anosov structures for plane translations was originally shown by W. White). P. Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. We prove that this claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time one map.