We study entropy and periodic-orbit growth for flows on metric spaces. First, we prove that the finite topological entropy of a flow on a compact metric space is completely carried by compact subsets of its regular set. Next, we establish a Bowen--Walters inequality for geometrically separating flows on possibly noncompact metric spaces, under uniform control at time zero and dynamical isolation at infinity. As applications, we obtain the Bowen--Walters inequality for singular suspension flows over expansive homeomorphisms, multisingular-hyperbolic sets---extending the upper-bound part of \cite{pyyz}---and asymptotically sectional-hyperbolic attractors such as Rovella's \cite{r}.
In this paper, we study random dynamical systems (RDSs) of homeomorphisms of the circle without a finite orbit. We characterize the topological dynamics of the associated semigroup by identifying the existence of invariant sets that are finite unions of intervals. We describe the accumulation points of the average or...
Dominique Malicet, Graccyela Salcedo· Ergodic Theory and Dynamical...· 0 citations
We study the singular sets of real-valued solutions to periodic parabolic equations with rapidly oscillating coefficients. Under a nondegeneracy condition on the corrector matrix, we obtain a local parabolic $n$-dimensional Hausdorff measure bound and a codimension-two Minkowski estimate. The Minkowski estimate holds a...
We study the Hausdorff dimension of attracting sets of three-dimensional flows whose one-dimensional quotient dynamics is non-uniformly expanding. Our main assumption is that the quotient map admits a full-branch uniformly expanding Gibbs--Markov induced map. We first prove that every such attractor contains a hyperbol...
We construct a smooth autonomous divergence-free vector field on $\mathbb{T}^3$ whose flow is exponentially mixing with respect to Lebesgue measure. This answers an open question about smooth autonomous exponential mixing in three dimensions. Starting from an exponentially mixing Anosov suspension flow, we perform surg...
We prove that a strongly nondegenerate Reeb vector field on a closed contact three-manifold has infinitely many simple positive hyperbolic periodic orbits if it has at least three simple periodic orbits. The main ingredient is a result for area-preserving maps on compact surfaces, possibly with boundary. We prove that,...
Masayuki Asaoka, Taisuke Shibata· 1 citation
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