The Dynamics of Measurable Pseudo-Anosov Maps

TL;DR

This paper investigates the dynamics of measurable pseudo-Anosov homeomorphisms on surfaces, revealing their transitivity, density of periodic points, ergodicity, and complex local structure despite lacking a global product structure.

Contribution

It introduces the concept of measurable pseudo-Anosov maps, extending Thurston's classical theory, and proves key dynamical properties for these generalized maps.

Findings

Measurable pseudo-Anosov maps are transitive.

They have dense periodic points.

They exhibit sensitive dependence on initial conditions.

Abstract

We study the dynamics of measurable pseudo-Anosov homeomorphisms of surfaces, a generalization of Thurston's pseudo-Anosov homeomorphisms. A measurable pseudo-Anosov map has a transverse pair of full measure turbulations consisting of streamlines which are dense immersed lines: these turbulations are equipped with measures which are expanded and contracted uniformly by the homeomorphism. The turbulations need not have a good product structure anywhere, but have some local structure imposed by the existence of tartans: bundles of unstable and stable streamline segments which intersect regularly, and on whose intersections the product of the measures on the turbulations agrees with the ambient measure. We prove that measurable pseudo-Anosov maps are transitive, have dense periodic points, sensitive dependence on initial conditions, and are ergodic with respect to the ambient measure. Measurable pseudo-Anosovs maps were introduced in [11], where we constructed a parameterized family of non-conjugate examples on the sphere.