Generalized foliations for homeomorphisms isotopic to a pseudo-Anosov homeomorphism: a geometric realization of a result by Fathi

TL;DR

This paper provides a new proof of Fathi's result, establishing that any surface homeomorphism isotopic to a pseudo-Anosov has associated stable and unstable invariant partitions resembling the classical foliations.

Contribution

It offers a geometric realization and a new proof of the invariant partition existence for homeomorphisms isotopic to pseudo-Anosov maps.

Findings

Constructs stable and unstable invariant partitions for such homeomorphisms

Shows these partitions share properties with pseudo-Anosov foliations

Provides a new geometric perspective on Fathi's result

Abstract

We give a new proof of a result by Fathi, which states that, to any homeomorphism of a closed surface which is isotopic to a pseudo-Anosov homeomorphism, we can associate a stable and an unstable invariant partition of the surface with properties which are similar to the unstable and the stable foliation of a pseudo-Anosov homeomorphism.