Lyapunov exponents and rigidity of Anosov automorphisms and skew products

Abstract

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L.We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

A note on rigidity of Anosov diffeomorphisms of the three torus

“…Previously with analogous hypothesis De la Llave in [8], proved the C 1 −conjugacy for Anosov diffeomorphisms on T 2 , and showed that the coincidence of periodic data is not sufficient to get C 1 conjugacy for Anosov diffeomorphisms on T 4 . During preparation of this work Saghin and Yang [15] announced also some rigidity results and in particular they prove that if the Lyapunov exponents (stable, weak unstable and strong unstable) of Lebesgue almost every point coincide with the exponents of linearization, then there is a smooth conjugacy. Our result is similar in spirit to theirs, although it is announced under different hypothesis and implies their result as a corollary (it does not mean necessarily being stronger).…”

“…[15], Theorem A). Let f be a C r , r ≥ 2, a volume preserving Anosov map such that TT3 = E s f ⊕ E wu f ⊕ E su f ,such that λ σ f (x) = λ σ A , σ ∈ {s, wu, su} for m− a.e.…”

Classification of partially hyperbolic diffeomorphisms under some rigid conditions

“…For perturbations of the linear Anosov map, the same result may also be obtained by using the first theorem in [SY19] once it is shown that the exponents of the Anosov diffeomorphism and its linear part are the same, which can be deduced from quasi-isometry of the foliations. A different approach to prove smooth conjugacy to a linear Anosov model was developed in [Var18] that uses smoothness of the center foliation plus extra requirements about the stable/unstable holonomies; to apply that approach one may have to establish that the hypotheses of our main theorem imply the requirements of [Var18], which does not seem to be direct.…”

Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

“…M. Poletti [11] used this notions in relation to geometric growth to prove a C 1 conjugacy result for Anosov maps of the 3-torus. Also very recently R. Saghin and J. Yang [14] used the entropy along expanding foliations as a tool to obtain Gibbs property of certain measures and then applied it to establish a local rigidity result of linear Anosov diffeomorphisms in terms of its Lyapunov exponents.…”