arXiv subjects
Publications and source records attributed to Marisa Cantarino.
We present computer-assisted proofs of partial hyperbolicity, existence of a blender and robust transitivity for diffeomorphisms on closed manifolds. These proofs are implemented for a family of derived-from-Anosov systems on the 3-torus.
We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic $u$-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special (has only one unstable direction per point), or the measure is the unique absolutely continuous invariant measure. We can apply this result in many settings, in particular obtaining uniqueness of $u$-Gibbs measures for every non-special perturbation of irreducible linear expanding maps of the torus with simple spectrum. This gives new open sets of partially hyperbolic systems displaying a unique $u$-Gibbs measure.
We consider the set of Hölder continuous cocycles over a finite shift acting on a group of Lipschitz homeomorphisms Lip(G), where G is a metrisable compact topological group. We establish that two dominated cocycles that coincide over periodic points of the base are cohomologous, with the conjugacy being Hölder continuous. Moreover, we prove that, under an additional condition, the existence of a measurable conjugacy implies the existence of a Hölder continuous conjugacy between the cocycles.
We prove the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundle. We also prove, regarding a class of potentials, the uniqueness of such measures for endomorphisms defined on the 2-torus that: have a linear model as a factor; and with the condition that this measure gives zero weight to the set where the conjugacy with the linear model fails to be invertible. In particular, we obtain the uniqueness of the measure of maximal entropy. For the n-torus, the uniqueness in the case with one-dimensional center holds for absolutely partially hyperbolic maps with additional hypotheses on the invariant leaves, namely, dynamical coherence and quasi-isometry.
We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the 2-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterization of smooth conjugacy between special Anosov endomorphisms and their linearizations.