Local rigidity of Lyapunov spectrum for toral automorphisms
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LOCAL RIGIDITY OF LYAPUNOV SPECTRUM FOR TORAL AUTOMORPHISMS BY
Andrey Gogolev∗ Department of Mathematics, Ohio State University, Columbus, OH 43210, USA e-mail: [email protected] AND
Boris Kalinin
∗∗
and Victoria Sadovskaya†
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA e-mail: [email protected], [email protected]
ABSTRACT
We study the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L4 is irreducible over Q. We consider a volume-preserving C 1 -small perturbation f of L. We show that if Lyapunov exponents of f with respect to the volume are the same as Lyapunov exponents of L, then f is C 1+H¨older conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.
1. Introduction and statements of results Hyperbolic and partially hyperbolic dynamical systems have been one of the main objects of study in smooth dynamics. Anosov automorphisms of tori are the prime examples of hyperbolic systems. The action of a hyperbolic ∗ Supported in part by NSF grant DMS-1823150. ∗∗ Supported in part by Simons Foundation grant 426243. † Supported in part by NSF grant DMS-1764216.
Received August 23, 2018
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A. GOGOLEV, B. KALININ AND V. SADOVSKAYA
Isr. J. Math.
matrix L ∈ SL(d, Z) on Rd induces an automorphism of the torus Td = Rd /Zd . It is well-known that any diffeomorphism f which is sufficiently C 1 close to L is also Anosov and topologically conjugate to L, i.e., there is a homeomorphism h of Td such that L = h−1 ◦ f ◦ h. The conjugacy h is unique in the homotopy class of the identity. It is only H¨older continuous in general, and various sufficient conditions for h to be smooth have been studied. For Anosov systems with one-dimensional stable and unstable distributions coincidence of eigenvalues of the derivatives of return maps at the corresponding periodic points was shown to imply smoothness of h [dlL87, dlLM88, dlL92, P90]. For higher dimensional systems even conjugacy of the derivatives of the corresponding return maps is not sufficient in general [dlL92, dlL02]. Nonetheless, smoothness of the conjugacy was established in higher dimensions under various additional assumptions [dlL02, KS03, dlL04, KS09, G08, GKS11]. The last paper establishes local rigidity for the most general class of hyperbolic automorphisms. Theorem ([GKS11, Theorem 1.1]): Let L : Td → Td be an irreducible Anosov automorphism such that no three of its eigenvalues have the same modulus. Let f be a C 1 -small perturbation of L such that the derivative Dp f n is conjugate to Ln whenever p = f n p. Then f is C 1+H¨older conjugate to L. We recall that a toral automorphism L is irreducible if it has no rational invariant subspaces, or equivalently if its characteristic polynomial is irreducible over Q. It follows that all eigenvalues of L are simple. Recently, R. Saghin and J. Yang obtained smoothne
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