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

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 W. White [Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 667–670 (1973; Zbl 0285.58009)]). P. Mendes [Proc. Am. Math. Soc. 63, 231–235 (1977; Zbl 0327.58009)] conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, S. Matsumoto [Ergodic Theory Dyn. Syst. 41, No. 3, 923–934 (2021; Zbl 1487.37035)] 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. For Part I, see [the authors, ibid. 35, No. 4, 1229–1242 (2015; Zbl 1355.37049)].