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Transitivity of surface dynamics lifted to Abelian covers

Published online by Cambridge University Press:  03 February 2009

PHILIP BOYLAND
PHILIP BOYLAND*
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA
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Abstract

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A homeomorphism f of a manifold M is called H1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover $\tilde {M}$. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H1(M). For a rel pseudo-Anosov map ϕ of a compact surface M we show that the following are equivalent: (a) ϕ is H1-transitive, (b) the action of ϕ on H1(M) has spectral radius one and (c) the lifts of the invariant foliations of ϕ to $\tilde {M}$ have dense leaves. The proof relies on a characterization of transitivity for twisted ℤd-extensions of a transitive subshift of finite type.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009, pp. 1417 - 1449
Copyright
Copyright © Cambridge University Press 2009

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