Partially volume expanding diffeomorphisms
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Annales Henri Poincar´ e
Partially volume expanding diffeomorphisms Shaobo Gan, Ming Li, Marcelo Viana
and Jiagang Yang
Abstract. We call a partially hyperbolic diffeomorphism partially volume expanding if the Jacobian restricted to any hyperplane that contains the unstable bundle E u is larger than 1. This is a C 1 open property. We show that any C 1+ partially volume expanding diffeomorphisms admits finitely many physical measures, and the union of their basins has full volume.
Contents 1. 2.
Introduction Preliminaries 2.1. Partial volume expansion 2.2. Gibbs u-states 3. A physical property 4. Proof of Theorem A 5. Proof of Theorem B 6. Partially volume expanding attracting sets 6.1. A semi-local finiteness theorem 6.2. A partially volume expanding solenoid References
1. Introduction Let f : M → M be a diffeomorphism on some compact Riemannian manifold M . An invariant probability μ of f is a physical measure if the set of points S.G. was supported by NSFC 11231001 and NSFC 11771025. M.L. was supported by NSFC 11571188 and NSFC 11971246. M.V. and J.Y. were partially supported by CNPq, FAPERJ, and PRONEX. This work was supported by the Fondation Louis D-Institut de France (project coordinated by M. Viana).
S. Gan et al.
Ann. Henri Poincar´e
z ∈ M for which n−1 1 ∗ δ i → μ (in the weak sense) n i=0 f (x)
(1)
has positive volume. This set is denoted by B(μ) and called the basin of μ. In the present paper, we investigate the existence and finiteness of physical measures in the setting of partially hyperbolic diffeomorphisms. More precisely, we assume that there exists a splitting T M = E u ⊕ E cs of the tangent bundle that is invariant under the tangent map Df and satisfies −1 −1 Df |Exu ) < 1 and Df |Exu )Df |Excs < 1 at every x ∈ M. (2) In other words, the unstable bundle E u is uniformly expanding, and it dominates the center-stable bundle E cs . A program for investigating the physical measures of partially hyperbolic diffeomorphisms was initiated by Alves, Bonatti, Viana in [1,8]. Their starting point was the observation that physical measures must be Gibbs u-states, a notion they borrowed from Pesin, Sinai [18] and whose definition we recall in Sect. 2.2. Not all Gibbs u-states are physical measures, but that is easily seen to be the case for the ergodic Gibbs u-states whose center Lyapunov exponents are all negative. Bonatti, Viana [8] introduced the notion of mostly contracting center, and they proved that under this condition there are finitely many ergodic Gibbs u-states, all their center Lyapunov exponents are negative, and they are the physical measures. Moreover, the union of their basins is a full-volume set. Andersson [2] later obtained an equivalent formulation which is the one we use in the present paper: Definition 1.1. A partially hyperbolic diffeomorphism has mostly contracting center if the Lyapunov exponents of every ergodic Gibbs u-state along the sub-bundle E cs are all negative. Further results on physical measures of diffeomorphisms with mostly contracting center have been obtain
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