On the ergodicity of partially hyperbolic systems
By Keith Burns, Amie Wilkinson. 2005
234 citations recorded by the source corpus.
Pugh and Shub have conjectured that essential accessibility implies ergodicity for a C , partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C 1Ci partially hyperbolic systems.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/451028422dd92ff43459f5dcaa6b942d964e1bb3
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Citations in the corpus
- Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms (2010, unknown): “For more references see [17].”
- New criteria for ergodicity and nonuniform hyperbolicity (2009, methodology): “Many of the arguments of [17] and [32] seem hard to generalize.”
- Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, background): “More detailed discussions can be found in [8], [16] and [17].”
- On the regularization of conservative maps (2008, methodology): “By construction, we get inductively that volF(l)(W )∩B(p, 1) = volW ∩ B(p, 1) for p ∈ {p1, ..., pl}, so it is clear that volF (s(W )) ∩ B(p, 1) = volW ∩ B(p, 1) except possibly for p = p2n−k .”
- Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle (2006, background): “Also, Burns and Wilkinson proved C r density of stable accessibility among skew products in [ 11 ].”
- Physical measures and absolute continuity for one-dimensional center direction (2013, background): “The first step is to note that f is ergodic, since it is accessible (see [28,37,50]).”
- Entropy along expanding foliations (2016, methodology): “The following result is a special case of a general result of Burns, Wilkinson [15]: Proposition 2.”
- A survey on partially hyperbolic dynamics (2006, methodology): “eorem 5.1 the C1 hypothesis was only used in the proof of proposition 5.3. In fact, this is the proposition that is being generalized for more general settings in [GrPuSh], [PuSh2], [PuSh3], [BuWi2], [BuWi3], [RHRHUr1]. Nevertheless, it is not clear that proposition 5.3 remains valid in the general ”
- Extremal Lyapunov exponents: an invariance principle and applications (2010, methodology): “Using that the center eigenvalues of A have norm 1, one easily gets that f is center bunched in the sense of [3, 18]: for some choice of the Riemannian structure”
- Accessibility (2018, unknown): “As highlighted by the Pugh–Shub Stable Ergodicity Conjecture, accessibility serves as a powerful tool to establish ergodicity—a result proven for systems with one-dimensional center bundles [HHU08a] and under center-bunched conditions [BW10].”
- The Entropy Conjecture for Diffeomorphisms away from Tangencies (2010, methodology): “Following Burns, Wilkinson [7] we avoid assuming dynamical coherence by using locally invariant (“fake”) foliations, a construction that goes back to Hirsch, Pugh, Shub [14].”