An infinite surface with the lattice property II: Dynamics of pseudo-Anosovs

TL;DR

This paper investigates the dynamics of hyperbolic affine automorphisms on an infinite-area translation surface, revealing non-recurrence, polynomial decay in cylinder actions, and detailed asymptotics of their effects on curves and homology.

Contribution

It provides new insights into the behavior of pseudo-Anosov-like automorphisms on infinite surfaces, including exact asymptotics and decay rates, extending classical finite-area theories.

Findings

Hyperbolic automorphisms are not recurrent.

Cylinder actions satisfy a polynomial decay mixing formula.

Automorphisms exhibit exponential attraction on curves and homology with polynomial decay.

Abstract

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. These objects are exponentially attracted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is attuned to the geometry of the surface and its deformations.