SRB measures for partially hyperbolic systems whose central direction is mostly expanding

By José F. Alves, C. Bonatti, Marcelo Viana. 2000

464 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6ebaec099acc673a9d55cb7c3a10c613dbe0fcb7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Entropy along expanding foliations (2016, methodology): “The proofs for the following basic properties of Gibbs u-states can be found in Bonatti, Dı́az and Viana [9, Subsection 11.2]: Proposition 5.1.”
Every compact manifold carries a completely hyperbolic diffeomorphism (2002, methodology): “Finally, let us mention that open sets of hyperbolic Bernoulli dieomorphisms on some manifolds are constructed in [ABV, BV , D, SW].”
Extremal Lyapunov exponents: an invariance principle and applications (2010, methodology): “The Invariance Principle allows them to treat the case of vanishing center Lyapunov exponent, which was not covered by [2, 15].”
SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, unknown): “Indeed, notice that there are no uniqueness statements in [4] where a different strategy is used for the proof of ﬁniteness.”
A survey on partially hyperbolic dynamics (2006, background): “• Examples that are not partially hyperbolic but presenting some form of weak hyperbolicity (see [ BoVi ]).”
Periodic points and homoclinic classes (2006, background): “But a homoclinic class may fail to be uniformly hyperbolic (for instance, it may contain in a robust way hyperbolic saddles having different indices as p, see the constructions in [D´ i1 ,D ´i2, DR] and the examples of non-hyperbolic robustly transitive diffeomorphisms in [BD1, BV ]) and locally ma”
Stable ergodicity of certain linear automorphisms of the torus (2002, methodology): “See for instance the examples of Bonatti and Viana [ BV ] where there the system has a dominated splitting with an expanding invariant bundle or the one studied in [T] where no invariant hyperbolic subbundle is available.”
Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps (2008, background): “In this paper we carry out the program set by Alves, Bonatti, Viana towards a theory of equilibrium states for the class of non-uniformly expanding maps originally proposed in [3, Appendix].”
SRB measures for non-hyperbolic systems with multidimensional expansion (2000, background): “Indeed, an application of these ideas is being given in [2], where partially hyperbolic systems are considered.”
Physical measures and absolute continuity for one-dimensional center direction (2013, methodology): “Following Pesin and Sinai [45] and Alves, Bonatti and Viana [6,21] (see also [19, Chapter 11]), we call Gibbs u-state any invariant probability measure m whose conditional probabilities (Rokhlin [52]) along strong unstable leaves are absolutely continuous with respect to the volume measure volu on t”
Lyapunov exponents with multiplicity 1 for deterministic products of matrices (2004, background): “exponents often permit a very precise description of the dynamics at the ergodic level (Sinai–Ruelle–Bowen measures) [1, 12]; and the study of certain transversely projective foliations, where non-zero exponents imply the uniqueness of the harmonic measure on the leaves [10, 11].”