Foliations and Conjugacy II: The Mendes Conjecture for Time-One Maps of\n Flows

2018/12/11 by Jorge Groisman, Groisman, Jorge, and Zbigniew Nitecki +1 · 1 citation
Mathematics · #37E30 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1812.04689

openalex publication_date 2018/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A diffeomorphism f:\ℝ²\→\ℝ² in the plane is Anosov if it\nhas a hyperbolic splitting at every point of the plane. The two known\ntopological conjugacy classes of such diffeomorphisms are linear hyperbolic\nautomorphisms and translations (the existence of Anosov structures for plane\ntranslations was originally shown by W. White). P. Mendes conjectured that\nthese are the only topological conjugacy classes for Anosov diffeomorphisms in\nthe plane. We prove that this claim holds when the Anosov diffeomorphism is the\ntime-one map of a flow, via a theorem about foliations invariant under a time\none map.\n

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