Nonuniform hyperbolicity for C 1-generic diffeomorphisms

Abstract

We study the ergodic theory of non-conservative C 1 -generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C 1 -generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C 1 -generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.In addition, confirming a claim made by R. Mañé in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C 1 .

Physical measures of discretizations of generic diffeomorphisms

“…We will use the connecting lemma for pseudo-orbits (see [BC04]), together with an ergodic closing lemma (adapted from [ABC11]) and the results of the appendix on the fact that the asymptotic rate is null (in particular Lemma 13) to prove that any invariant measure of the diffeomorphism can be observed by starting at any point of a generic subset of T n .…”

“…Let λ be the smallest absolute value of the Lyapunov exponents of µ (in particular, λ > 0). We choose a point x which is regular for the measure µ: we suppose that it satisfies the conclusions of Oseledets' and Birkhoff's theorems, and Mañé's ergodic closing lemma (see [ABC11,paragraph 6.1]). We denote by F f x the stable subspace and G f x the unstable subspace for the Oseledets' splitting at the point x.…”

“…This will allow us to merge some orbits of the discretizations. The proof of Theorem A also uses two connecting lemmas (the connecting lemma for pseudo-orbits of [BC04] and an improvement of the ergodic closing lemma of [ABC11]) and a statement of local linearization of a C 1 -diffeomorphism (Lemma 23) involving the regularization result due to Avila [Avi10].…”

Usual limit shadowable homoclinic classes of generic diffeomorphisms

“…In this regards, we mention some involving results. In [11,12], Crovisier explained Mañé's ergodic closing lemma which gives the measure theoretical viewpoint on the approximation by periodic orbit, that is, any ergodic invariant probability measure μ of Moreover, the orbit O f (p n ) converges to the support of μ in the Hausdor topology.…”

The Pesin entropy formula for diffeomorphisms with dominated splitting

“…To find new results that hold for C 1 maps relatively recent research started assuming some uniformly dominated conditions (see [ABC11,BCS13,ST10,ST12,T02]). …”