Transitivity of Surface Dynamics Lifted to Abelian Covers.pdfVIP

Transitivity of Surface Dynamics Lifted to Abelian Covers Philip Boyland Dept. of Mathematics University of Florida Gainesville, FL 326118105 Abstract: A homeomorphism f of a manifold M is called H transitive if there is a tran 1 8 ˜ sitive lift of an iterate of f to the universal Abelian cover M . Roughly speaking, this means 0 that f has orbits which repeatedly and densely explore all elements of H (M). For a rel 0 1 2 pseudoAnosov map φ of a compact surface M we show that the following are equivalent: r (a) φ is H transitive, (b) the action of φ on H (M) has spectral radius one, and (c) the lifts p 1 ˜ 1 A of the invariant foliations of φ to M have dense leaves. The proof relies on a characterization d 4 of transitivity for twisted Z extensions of a transitive subshift of finite type. 1 ] 1 Introduction S D There are many ways to characterize the complexity of a dynamical system on a manifold . M . In this paper we focus on the characterization of H -transitivity. A homeomorphism f h 1 t is called H transitive when, roughly speaking, it has orbits which repeatedly and densely a 1 m explore all the elements of first homology. As is natural and commonly done, we formalize [