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

By C. Bonatti, Marcelo Viana. 2000

361 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dc3afd00588ab1324bcb5c90d10457531bdcf8c3

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Citations in the corpus

Physical measures and absolute continuity for one-dimensional center direction (2013, methodology): “The following fact was first observed by Bonatti and Viana [21]:”
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].”
SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, neutral): “The conclusions of Corollary B are obtained in [13] in the quite distinct setting of a partially hyperbolic diffeomorphism with an invariant tangent bundle decomposition E cs ⊕ E u (i.e. with uniform expansion and nonuniform contraction, see Section 2 for an in-depth historical literature review of…”
Nonuniform hyperbolicity for C^1-generic diffeomorphisms (2008, background): “Λ. There are examples of nonhyperbolic robustly transitive sets – and hence of isolated transitive sets of C1-generic diﬀeomorphisms – whose ﬁnest dominated splitting has only two bundles Eand F, see [BonV]. Thus in such examples all of the generic measures provided by Proposition 7.4 above must hav”
Stably ergodic diffeomorphisms which are not partially hyperbolic (2002, neutral): “(For an example of such vector fields see [BV00]) More precisely, first we modify along stable direction E i (pi) ⊕ E i+1(pi) for 1 ≤ i ≤ n − 3 until the index of pi drops one and two fixed points qi, ri are created.”
A survey on partially hyperbolic dynamics (2006, methodology): “ manifold (see [BoD´ı]). • Perturbations of the time-one map of a transitive Anosov ﬂow (see [BoD´ı]). • Examples that are not partially hyperbolic but presenting some form of weak hyperbolicity (see [BoVi]). An important tool in order to construct examples are the center-stable blenders ﬁrst introd”
On the regularization of conservative maps (2008, background): “The existence of locally generic non-uniformly hyperbolic ergodic conservative diffeomorphisms with non-simple Lyapunov spectrum [11], [26] (the proof, conditional to the existence of regularization, is nicely sketched in page 260 of [8]).”
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].”
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.”
Partial hyperbolicity for symplectic diffeomorphisms (2004, background): “In compact manifolds of dimension greater than two, robust transitivity does not imply uniform hyperbolicity, see [14,10,4,6,15].”