SRB measures for attractors with continuous invariant splittings

Mi, Zeya; Cao, Yongluo; Yang, Dinghui

doi:10.1007/s00209-017-1883-2

Cited by 5 publications

(2 citation statements)

References 22 publications

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“…The following version of Pliss lemma helps us to select good points with sufficient hyperbolic property. See [26,Lemma 2.1] for a proof. The following result provides a periodic approximation of ergodic physical measures.…”

Section: 24mentioning

confidence: 99%

SRB measures for partially hyperbolic flows with mostly expanding center

Mi¹,

You²,

Zang³

2020

Preprint

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We prove that a partially hyperbolic attractor for a C 1 vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is C 2 , and the center bundle admits the sectional expanding condition w.r.t. any Gibbs u-state, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.

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Section: 24mentioning

confidence: 99%

SRB measures for partially hyperbolic flows with mostly expanding center

Mi¹,

You²,

Zang³

2020

Preprint

Self Cite

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“…As for the construction of SRB measures beyond the setting of uniform hyperbolicity one should mention the construction of u-Gibbs measures for partially hyperbolic attractors that admit an unstable bundle by Pesin and Sinai [34], the existence and uniqueness of the SRB measure for the class of robustly transitive diffeomorphisms derived from Anosov constructed by Mañé by Carvalho [12], and the construction of SRB measures for C 1+α partially hyperbolic diffeomorphisms displaying a non-uniform hyperbolicity condition along the central direction by Alves, Bonatti and Viana [1,11]. More recently, Mi, Cao and Yang [30] constructed SRB measures for attractors of C 2 diffeomorphisms that admit Hölder continuous invariant (non-dominated) splittings with some non-uniform expansion. After Young's [51] axiomatic construction for studying decay of correlations, the decay of correlations for SRB measures can be studied through the existence of Markov towers and was shown to depend on the Lebesgue measure of the tails associated to non-uniform hyperbolicity.…”

Section: Introductionmentioning

confidence: 99%

SRB measures for partially hyperbolic attractors of local diffeomorphisms

Cruz

Varandas

2018

Ergod. Th. Dynam. Sys.

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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 ...

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SRB Measures for Partially Hyperbolic Flows with Mostly Expanding Center

You

Zang

2022

J Dyn Diff Equat

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SRB measures for attractors with continuous invariant splittings

Cited by 5 publications

References 22 publications

SRB measures for partially hyperbolic flows with mostly expanding center

SRB measures for partially hyperbolic flows with mostly expanding center

SRB measures for partially hyperbolic attractors of local diffeomorphisms

SRB Measures for Partially Hyperbolic Flows with Mostly Expanding Center

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