scholarly journals SRB measures for partially hyperbolic attractors of local diffeomorphisms

2018 ◽  
Vol 40 (6) ◽  
pp. 1545-1593
Author(s):  
ANDERSON CRUZ ◽  
PAULO VARANDAS
Keyword(s):  
Thermodynamic Formalism ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Local Diffeomorphism ◽  
Local Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Cone Field ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic

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$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

Volume lemmas and large deviations for partially hyperbolic endomorphisms

2019 ◽  
Vol 41 (1) ◽  
pp. 213-240
Author(s):  
ANDERSON CRUZ ◽  
GIOVANE FERREIRA ◽  
PAULO VARANDAS
Keyword(s):  
Lebesgue Measure ◽  
Large Deviation ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Srb Measure ◽  
Cone Field ◽  
Mild Assumption ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Set Of Points

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence, under a mild assumption we prove exponential large-deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms

1993 ◽  
Vol 13 (1) ◽  
pp. 21-44 ◽  
Author(s):  
Maria Carvalho
Keyword(s):  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Saddle Node ◽  
Basic Set ◽  
Hyperbolic Attractors ◽  
Generic Bifurcation

AbstractThis paper is about the existence of transitive non-hyperbolic attractors with corresponding SRB measures for arcs of diffeomorphisms crossing the boundary of the Axiom A systems, obtained through an elementary generic bifurcation (Hopf, saddle-node or flip) on a transitive Anosov diffeomorphism or an attracting basic set.

scholarly journals Almost sure rates of mixing for partially hyperbolic attractors

2022 ◽  
Vol 311 ◽  
pp. 98-157
Author(s):  
José F. Alves ◽  
Wael Bahsoun ◽  
Marks Ruziboev
Keyword(s):  
Hyperbolic Attractors ◽  
Partially Hyperbolic

scholarly journals Equilibrium states for a class of skew products

2019 ◽  
Vol 40 (11) ◽  
pp. 3030-3050
Author(s):  
MARIA CARVALHO ◽  
SEBASTIÁN A. PÉREZ
Keyword(s):  
Existence And Uniqueness ◽  
Global Analysis ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Axiom A ◽  
Skew Products ◽  
Pure Mathematics ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We consider skew products on $M\times \mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

scholarly journals Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers

2004 ◽  
Vol 160 (2) ◽  
pp. 375-432 ◽  
Author(s):  
Carlos Morales Rojas ◽  
Maria Pacifico ◽  
Enrique Pujals
Keyword(s):  
Singular Sets ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic

scholarly journals Gibbs measures for partially hyperbolic attractors

1982 ◽  
Vol 2 (3-4) ◽  
pp. 417-438 ◽  
Author(s):  
Ya. B. Pesin ◽  
Ya. G. Sinai
Keyword(s):  
Special Form ◽  
Gibbs Measures ◽  
Hyperbolic Attractor ◽  
Absolutely Continuous ◽  
Unstable Manifolds ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Limit Points ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

AbstractWe consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.

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