$\mathcal{F}$-expansivity for Borel measures

Abstract

We introduce the notion of $\mathcal{F}$-expansive measure by making the dynamical ball in [4] to depend on a given subset $\mathcal{F}$ of the set of all the reparametrizations $\mathcal{H}$. We prove that these measures satisfy some interesting properties resembling the expansive ones. These include the equivalence with expansivity when $\mathcal{F}=\mathcal{H}$, the vanishing along the orbits, the absence of singularities in the support, the $\mathcal{F}$-expansivity with respect to time t-maps, the invariance under equivalence and the characterization for suspensions. We also analyze the support of the $\mathcal{F}$-expansive measures and prove that there exists a dense subset of measures (in the set of $\mathcal{F}$-expansive measures) all of them with a common support. Finally, we extend to flows the recent result for homeomorphisms in [11].