We show that a self orbit equivalence of a transitive Anosov flow on a 3-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is R-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev [FG16] about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is 3-dimensional.
Let L be a hyperbolic automorphism of T d , d ≥ 3. We study the smooth conjugacy problem in a small C 1 -neighborhood U of L.The main result establishes C 1+ν regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are C 1 -close to an irreducible linear hyperbolic automorphism L with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations.We elaborate on the example of de la Llave of two Anosov systems on T 4 with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb C 1+ν conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of T 4 C 1 -close to the original example.
We consider an irreducible Anosov automorphism L of a torus T d such that no three eigenvalues have the same modulus. We show that L is locally rigid, that is, L is C 1+Hölder conjugate to any C 1 -small perturbation f such that the derivative D p f n is conjugate to L n whenever f n p = p. We also prove that toral automorphisms satisfying these assumptions are generic in SL(d, Z). Examples constructed in the Appendix show importance of the assumption on the eigenvalues. F. Voloch. Unit in a number field with same at a real and a complex place. URL:
We show that for a large space of volume preserving partially hyperbolic diffeomorphisms of the 3-torus with non-compact central leaves the central foliation generically is non-absolutely continuous.
Abstract. We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms f on compact 3-manifolds with fundamental groups of exponential growth such that f n is not homotopic to identity for all n > 0. These provide counterexamples to a classification conjecture of Pujals.
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