Measure rigidity for leafwise weakly rigid actions

In this paper, given a Borel action $G\curvearrowright X$, we introduce a new approach to obtain classification of conditional measures along a $G$-invariant foliation along which $G$ has a controlled behavior. Given a Borel action $G\curvearrowright X$ over a Lebesgue space $X$ we show that if $G\curvearrowright X$ preserves an invariant system of metrics along a Borel lamination $\mathcal F$, which satisfy a good packing estimative hypothesis, then the ergodic measures preserved by the action are rigid in the sense that the system of conditional measures with respect to the partition $\mathcal F$ are the Hausdorff measures given by the metric system or are supported in a countable number of boundaries of balls. The argument we employ does not require any structure on $G$ other then second-countability and no hyperbolicity on the action as well. Our main result is interesting on its own, but to exemplify its strength and usefulness we show some applications in the context of cocycles over hyperbolic maps and to certain partially hyperbolic maps

Anosov Endomorphisms on the 2-torus: Regularity of foliations and rigidity

We give on the 2-torus a characterization of smooth conjugacy of special Anosov endomorphisms with their linearizations in terms of the regularity of the stable and unstable foliations. This regularity condition is the uniform bounded density (UBD) property, which is the uniform version absolute continuity for foliations

There exist transitive piecewise smooth vector fields on S2\mathbb{S}^2 but not robustly transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $\S^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $\S^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $\S^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $\S^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds

The Orbit Space Approach for Piecewise Smooth Vector Fields

In this work we establish a well defined theory of 'orbit spaces' for nonsmooth vector fields (NSVF). This approach is inspired from the techniques already used in the study of endomorphism, namely inverse limit analysis. We then apply the construction of our theory for the understanding of transitivity in NSVF. We also prove that the known examples of transitive NSVF are indeed transitive in the 'orbit space' as a consequence of our general theorem of transitivity

Equilibrium states for maps isotopic to Anosov

In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on T4, with two-dimensional center foliation. To do so we propose to study the disintegration of measures along one-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context.Fundação de Amparo à Pesquisa do Estado de São Paulo/[2018/18990-0]/FAPESP/BrasilCoordinación de la formación del personal de nivel superior/[2019/88882.329056-01]/CAPES/BrasilFundação de Amparo à Pesquisa do Estado de São Paulo/[17/06463-3]/FAPESP/BrasilFundação de Amparo à Pesquisa do Estado de São Paulo/[16/22475-9]/FAPESP/BrasilUniversidad de Costa Rica/[]/UCR/Costa RicaUCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de MatemáticaUCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA