Related papers: Entropy along expanding foliations
Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $$\lambda_{s}<1<\lambda_{c}<\lambda_{u}.$$ Under…
We prove that every C1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub's entropy conjecture: the entropy is bounded from below…
Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$ where $\lambda^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an…
In this article we study the regularity of the topological and metric entropy of partially hyperbolic flows with two-dimensional center direction. We show that the topological entropy is upper semicontinuous with respect to the flow, and we…
We describe dimensional entropies introduced in a previous work list some of their properties and give some new proofs. These entropies allowed the definition of entropy-expanding maps. We introduce a new notion of entropy-hyperbolicity for…
In this work, we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class…
We study the topological properties of expanding invariant foliations of $C^{1+}$ diffeomorphisms, in the context of partially hyperbolic diffeomorphisms and laminations with $1$-dimensional center bundle. In this first version of the…
In this paper, we provide an upper bound on the number of maximal entropy ergodic measures with zero Lyapunov exponent for topologically transitive partially hyperbolic diffeomorphisms with compact one-dimensional center leaves on…
We study the $C^1$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^2$ partially hyperbolic symplectic systems which have bounded $C^2$ distance to the identity. In this set, we prove…
We prove that generically in $\text{Diff}^{1}_{m}(M)$, if an expanding $f$-invariant foliation $W$ of dimension $u$ is minimal and there is a periodic point of unstable index $u$, the foliation is stably minimal. By this we mean there is a…
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are…
We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguy\^en and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted…
We show that for any $C^1$ partially hyperbolic diffeomorphism, there is a full volume subset, such that any Cesaro limit of any point in this subset satisfies the Pesin formula for partial entropy. This result has several important…
We prove that for $\mathcal{C}^{1,\alpha}$ diffeomorphisms on a compact manifold $M$ with ${\rm dim} M\leq 3$, if an invariant measure $\mu$ is a continuity point of the sum of positive Lyapunov exponents, then $\mu$ is an upper…
Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>\lambda_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$…
We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp…
We obtain new entropy rigidity results for $u$-Gibbs measures by showing that whenever a $u$-Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be…
In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism.…
In the theory of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, it is possible to compute a one-dimensional representative map for any irreducible isotopy class. The topological entropy of this graph representative…
We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular…