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Foliations and conjugacy, II: the Mendes conjecture for time-one maps of flows

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems

Published online by Cambridge University Press:  30 October 2020

JORGE GROISMAN  and
ZBIGNIEW NITECKI
JORGE GROISMAN
Affiliation:
Instituto de Matemática y Estadística Prof. Ing. Rafael Laguardia, Facultad de Ingeniería Julio Herrera y Reissig 565 11300, Montevideo, Uruguay (e-mail: jorgeg@fing.edu.uy)
ZBIGNIEW NITECKI*
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, USA
*
e-mail: zbigniew.nitecki@tufts.edu
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Abstract

A diffeomorphism of the plane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, Matsumoto gave an example of an Anosov diffeomorphism of the plane, which is a Brouwer translation but not topologically conjugate to a translation, disproving Mendes’ conjecture. In this paper we prove that Mendes’ claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time-one map. In particular, this shows that the kind of counterexample constructed by Matsumoto cannot be obtained from a flow on the plane.

Keywords

dynamics in the planeAnosov diffeomorphismsplanar flowsMendes conjecture

MSC classification

Primary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3307 - 3324
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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