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The dynamics of measurable pseudo-Anosov maps

Volume 267 / 2024

Philip Boyland, André de Carvalho, Toby Hall Fundamenta Mathematicae 267 (2024), 1-24 MSC: Primary 37E30; Secondary 57K20 DOI: 10.4064/fm240216-22-7 Published online: 10 September 2024

Abstract

We study the dynamics of measurable pseudo-Anosov homeomorphisms of surfaces, a generalization of Thurston’s pseudo-Anosov homeomorphisms. A measurable pseudo-Anosov map has a transverse pair of full measure turbulations consisting of streamlines which are dense immersed lines: these turbulations are equipped with measures which are expanded and contracted uniformly by the homeomorphism. The turbulations need not have a good product structure anywhere, but have some local structure imposed by the existence of tartans: bundles of unstable and stable streamline segments which intersect regularly, and on whose intersections the product of the measures on the turbulations agrees with the ambient measure.

We prove that measurable pseudo-Anosov maps are transitive, have dense periodic points, sensitive dependence on initial conditions, and are ergodic with respect to the ambient measure.

Measurable pseudo-Anosov maps were introduced in our earlier paper [arXiv:2306.16059], where we constructed a parameterized family of non-conjugate examples on the sphere.

Authors

  • Philip BoylandDepartment of Mathematics
    University of Florida
    Gainesville, FL 32611-8105, USA
    e-mail
  • André de CarvalhoDepartamento de Matemática Aplicada
    IME-USP
    05508-090 São Paulo SP, Brazil
    e-mail
  • Toby HallDepartment of Mathematical Sciences
    University of Liverpool
    Liverpool L69 7ZL, UK
    e-mail

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