Endperiodic Automorphisms of Surfaces and Foliations

TL;DR

This paper extends Handel and Miller's classification of endperiodic automorphisms of surfaces, introducing an axiomatic framework for geodesic laminations, and proves isotopy and transfer theorems relevant to surface automorphisms and 3-manifold foliations.

Contribution

It develops an axiomatic theory for pseudo-geodesic laminations, proves their isotopy to geodesic laminations, and establishes a transfer theorem for foliations with complex dynamics.

Findings

Geodesic laminations satisfy the axioms and are isotopic to pseudo-geodesic laminations.

Endperiodic automorphisms can be isotoped to smooth automorphisms preserving smooth laminations.

The transfer theorem links dynamics of foliations in 3-manifolds with Handel-Miller theory.

Abstract

We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel-Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic lamaniations, show the geodesic laminations satisfy the axioms, and prove that paeudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the "transfer theorem" for foliations of 3-manifolds., namely that, if two depth one foliations are transverse to a common one-dimensional foliation whose monodromy on the noncompact leaves of the first foliation exhibits the nice dynamics of Handel-Miller theory, then the transverse one-dimensional foliation also induces monodromy on the noncompact leaves of the second foliation exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.