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

W Patrick Hooper

doi:10.3934/jmd.2019009

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.

Highlights

Translation surfaces built from two copies of a regular polygon as depicted in Figure 1 were studied by Veech and proven to have beautiful properties [9]
A significant goal of this paper is to address the extent to which hyperbolic affine automorphisms of infinite surfaces satisfy the defining properties of pseudo-Anosov homeomorphisms of closed surfaces
We show that the projectivized classes P and P respectively attract and repel every element of P ◦ ∩∗ H1(P◦1; R), but unlike in prior results the rate of polynomial decay depends on the chosen homology class

Summary

INTRODUCTION

Translation surfaces built from two copies of a regular polygon as depicted in Figure 1 were studied by Veech and proven to have beautiful properties [9]. Letting i : S × S → R denote geometric intersection number (i.e., the minimum number of transverse intersections among curves from the isotopy classes), we have an induced map i∗ : S → RS defined by i∗(α)(β) = i (α, β) so that the image is contained in the non-negative cone of RS. (λ1s )−n φ−n(γ) converges to a multiple of μs, but the rate given by j is determined by a different sequence of nested subspaces This theorem is a consequence of our Theorem 19 which is stronger in that it gives a formula for the constant appearing in the limit in statement (3). Fixing a curve γ ∈ P1 representing a homology class we can use these canonical homeomorphisms to obtain corresponding curves in Pc. We observe that the holonomies of these curves measured on Pc denoted holc γ ∈ R2 depend polynomially in c. The key is the observation that the area of intersection of two cylinders is largely governed by algebraic intersection numbers between the core curves. – In §3.7 we consider geometric intersection numbers and prove Theorem 3

BACKGROUND

A SUBGROUP DEFORMATION

THE PARABOLA SURFACE