Regularity of foliations and Lyapunov exponents of partially hyperbolic dynamics on 3-torus

Abstract

Abstract. In this work we study relations between regularity of invariant foliations and Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of desintegration of Lebesgue measure which can be considered as a criterium for rigidity of Lyapunov exponents.

Central Lyapunov exponent of partially hyperbolic diffeomorphisms of 𝕋³

“…We believe that it is not the case and formulate the following question. We remark that A. Tahzibi and F. Micena [11] recently gave an affirmative answer to this question assuming F c satisfies a uniformly bounded density condition which is a regularity condition stronger than leafwise absolute continuity.…”

A note on rigidity of Anosov diffeomorphisms of the three torus

“…Proof. If F wu has UBD property, F wu is absolutely continuous and by [11], λ wu f (x) = λ wu A , for m a.e. x ∈ T 3 .…”

“…We have defined the notion of foliations with Uniform Bounded Density in [11] as follows: Definition 1.1 ([11]). We say that a foliation F has Uniform Bounded Density property (or UBD for short) if there is a uniform constant K > 1, such that for any F −foliated box B (independent of the size of the box) In what follows we consider the derived from Anosov setting (partially hyperbolic with Anosov linearization) on T 3 .…”

Lyapunov exponents and smooth invariant foliations for partially hyperbolic diffeomorphisms on

“…Therefore the smoothness of the foliation gives a much stronger result. We point out that the methods from this paper are different from those used in [8].…”

“…See [7] for an example of a DA diffeomorphism with C 1 centre leaves, but which is not C 1 conjugate to its linearization, compare to Corollary 2.4 below. If we exclude the C 1 hypothesis from Theorem 2.2, then from [8] the Lyapunov exponents are constant volume almost everywhere and equal to the Lyapunov exponent of the linearization. Therefore the smoothness of the foliation gives a much stronger result.…”