SRB measures for partially hyperbolic attractors of local diffeomorphisms

Abstract

In the present paper we contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak * topology) and that their entropy varies continuously with respect to the local diffeomorphism.In parallel to the developments of partially hyperbolic diffeomorphisms, there have been important contributions to the study of (non-singular) endomorphisms, i.e., local diffeomorphisms. In the case that the endomorphisms admit no stable bundle, the geometrical constructions of Markov structures for endomorphisms with non-uniform expansion due to [3] provide weak conditions for the existence of SRB measures and estimates on their decay of correlations. The situation is substantially more complicated in the case of endomorphisms displaying contracting behavior, as e.g. strongly dissipative endomorphisms that arise in the context of bifurcation of homoclinic tangencies associated to periodic points of diffeomorphisms (see [33] and references therein). In the mid seventies, Przytycki [35] extended the notion of uniform hyperbolicity to the context of endomorphisms and studied Anosov endomorphisms. Here, due to the non-invertibility of the dynamics, the existence of an invariant unstable subbundle for uniformly hyperbolic basic pieces needs to be replaced by the existence of positively invariant cone-fields on which vectors are uniformly expanded by all positive iterates (we refer the reader to Subsection 4.3 for more details). In [38,39] the authors constructed SRB measures for Axiom A attractors of endomorphisms, obtaining these as equilibrium states for the geometric potential (defined by means of the natural extension). A characterization of SRB measures for uniformly hyperbolic endomorphisms can also be given in terms of dimensional characteristics of the stable manifold ([47]). The thermodynamic formalism of hyperbolic basic pieces for endomorphisms had the contribution of Mihailescu and Urbanski [31,32] that introduced and constructed inverse SRB measures for hyperbolic attractors of endomorphisms. Among the difficulties that arise when dealing with non-invertible hyperbolic dynamics one should refer the absence of contraction along inverse branches and the fact that unstable manifolds depend on entire pre-orbits, thus unstable manifolds may have a complicated geometrical structure (cf. [37,47]).The ergodic theory of partially hyperbolic endomorphisms is more incomplete. In the context of surface endomorphisms, a major contribution is due to Tsujii [46], which proved that for r ≥ 19, C r ...

Volume lemmas and large deviations for partially hyperbolic endomorphisms

“…The construction of the SRB measures for these partially hyperbolic attractors follows from careful control of densities of the pushforward of the Lebesgue measure on disks tangent to the cone field [7]. By (H3), there exists a disk D that is tangent to the cone field C and so that (2.1) holds for a positive Lebesgue measure set in D and SRB measures arise from the ergodic components of the accumulation points of the Cesàro averages…”

“…4. A first volume lemma 4.1. c-Cone-hyperbolic times and geometry of disks tangent to C. In this subsection we collect some results from [7], which were inspired by [1]. Consider D a disk in M which is tangent to the cone field C and assume that D satisfies Leb D (H ) > 0, where H is the subset given by hypothesis (H3).…”

“…Here we consider attractors for C 1+α (α > 0) non-singular endomorphisms on compact Riemannian manifolds that admit uniform contraction along a well-defined invariant stable subbundle and have a positively invariant cone field. Under a mild non-uniform cone-expansion assumption, these endomorphisms admit finitely many hyperbolic and physical SRB measures, these are statistically stable and their metric entropies vary continuously with respect to the underlying dynamics, and the SRB measure is unique if the attractor is transitive (here Volume lemmas and large deviations for partially hyperbolic maps 215 we mean SRB measures as measures which are absolutely continuous disintegrations with respect to Lebesgue measure along Pesin unstable manifolds) [7]. In this context, unstable subbundles depend on the pre-orbits and are almost everywhere defined via Oseledets's theorem, hence the unstable Jacobian is defined almost everywhere (with respect to the SRB measure) and could fail to be Hölder continuous on the attractor.…”

“…Our main contribution here is to overcome such lack of regularity of the unstable Jacobian. For this, we prove a bounded distortion property and obtain two volume lemmas at instants of hyperbolicity (c-cone-hyperbolic times defined in [7]). The (invariant) SRB measure is supported on the attractor, and the volume lemma for the SRB measure lies on the regularity of the unstable Jacobian along pre-images of unstable disks.…”

“…PROPOSITION 5.1. [7] Given y ∈ supp(ν), there existsx = (x −n ) n∈N ∈ M f such that y belongs to a disk (x) of radius δ > 0 accumulated by n j -hyperbolic disks ( f n j (x), δ) with j → ∞, and satisfying that:…”