Endperiodic Automorphisms of Surfaces and Foliations

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We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces.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.