Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle

By F. R. Hertz, M. R. Hertz, R. Ures. 2006

135 citations recorded by the source corpus.

We prove that stable ergodicity is Cr open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r∈[2,∞].The proof follows the Pugh–Shub program [29]: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is Cr open and dense, and essential accessibility implies ergodicity.

Subfields: Cell Biology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a69507e227ef055bde60142bcb69f080e3c33e0d

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Citations in the corpus

Leaf conjugacies on the torus (2013, background): “For generic f , however, E f ⊕ E u f is not integrable [14], so it does not make sense to talk of us-leaves of f .”
Entropy along expanding foliations (2016, background): “Moreover, the subset of accessible systems is C1 open and Ck dense for any k ≥ 1 among all partially hyperbolic diffeomorphisms with one-dimensional center direction, by Burns, Rodriguez Hertz, Rodriguez Hertz, Talitskaya and Ures [17, 14]; see also Theorem 1.5 in Niţică and Török [25].”
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]).”
Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, background): “More recently, Rodriguez Hertz, Rodriguez Hertz and Ures [47] checked the complete conjecture for conservative systems whose center bundle is onedimensional: accessibility is Cr -dense among Cr partially hyperbolic diffeomorphisms, for any r ≥ 1.”
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].”
New criteria for ergodicity and nonuniform hyperbolicity (2009, background): “3 of [32]) to obtain an open accessibility class perturbing f2 in an arbitrary small neighborhood of any fixed point of {q}×(0, 1 2).”