Publicaciones

We prove that if $f$ is a $C^{1+}$ partially hyperbolic diffeomorphism satisfying certain conditions then there is a $C^1$-open neighborhood $\mathcal{A}$ of $f$ so that every $g \in \mathcal{A}\cap \operatorname{Diff}^{1+}(M)$ has a unique equilibrium state.

We prove the finiteness of ergodic measures of maximal entropy for an open set of partially hyperbolic diffeomorphisms. We also study the upper semicontinuity of the number of measures of maximal entropy with respect to the diffeomorphism.

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute value of the Lyapunov exponents. As applications we prove finiteness for a class derived from Anosov partially hyperbolic diffeomorphisms defined on $\mathbb{T}^4$ and that in a class of skew product over partially hyperbolic diffeomorphisms there exists a $C^1$ open and $C^r$ dense set of diffeomorphisms with a finite number of ergodic measures of maximal entropy. We also study the upper semicontinuity of the number of measures of maximal entropy with respect to the diffeomorphism.

We construct symbolic dynamics for flows with positive speed in any dimension: for each $\chi > 0$, we code a set that has full measure for every invariant probability measure which is $\chi$-hyperbolic.