Pseudo-Anosov maps and continuum theory

TL;DR

This paper explores the dynamics of pseudo-Anosov maps within continuum theory, introducing a new metric space framework that links stable foliations to hypercontinua and generalizes key dynamical properties.

Contribution

It develops a novel compactification of the hyperspace of subcontinua using a second order Hausdorff distance, connecting stable foliations with hypercontinua and extending dynamical properties to cw-expansive maps.

Findings

Negative iterates of stable arcs converge to hypercontinua.

Stable leaves are dense and maps exhibit topological mixing.

The framework generalizes pseudo-Anosov dynamics to broader spaces.

Abstract

In the hyperspace of subcontinua of a compact surface we consider a second order Hausdorff distance. This metric space is compactified in such a way that the stable foliation of a pseudo-Anosov map is naturally identified with a hypercontinuum. We show that negative iterates of a stable arc converges to this hypercontinuum in the considered metric. Some dynamical properties of pseudo-Anosov maps, as topological mixing and the density of stable leaves, are generalized for cw-expansive homeomorphisms of pseudo-Anosov type on compact metric spaces.