Related papers: Entropy of semiclassical measures in dimension 2

We study the entropy of Sinai-Ruelle-Bowen measure of the geodesic flow on convex real projective surfaces, and shows that the Hilbert area tends to infinity if the entropy tends to zero. For the Blaschke metric, the area tends to infinity…

In this paper, we prove Ruelle's inequality for the geodesic flow in non-compact manifolds with Anosov geodesic flow and some assumptions on the curvature. In the same way, we obtain Pesin's formula for Anosov geodesic flow in non-compact…

Motivated by a paper of Bolsinov and Taimanov DG/9911193 we consider non-holonomic situation and exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Moreover the Riemannian examples…

We show the equivalences of several notions of entropy, like a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform…

In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the $C^0$ topology. We establish several instances of entropy robustness (persistence of entropy non-vanishing…

We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (Ebin) distance…

We prove that a closed surface with a CAT($\kappa$) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between…

We compare the regularity of the boundary of a convex set with the value of its Finslerian volume entropy. The main result states that the volume entropy of a two-dimensional domain whose associated curvature measure is Ahlfors…

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a…

We consider a Laplace eigenfunction $\varphi_\lambda$ on a smooth closed Riemannian manifold, that is, satisfying $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$. We introduce several observations about the geometry of its vanishing…

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible…

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the…

We contribute to an original problem studied by Hamilton and others, in order to understand the behaviour of maximal solutions of the Ricci flow both in compact and non-compact complete orientable Riemannian manifolds of finite volume. The…

We investigate typical behavior of geodesics on a closed flat surface $S$ of genus $g\geq 2$. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same…

Given a compact topological dynamical system (X, f) with positive entropy and upper semi-continuous entropy map, and any closed invariant subset $Y \subset X$ with positive entropy, we show that there exists a continuous roof function such…

We obtain a $C^1$-generic subset of the incompressible flows in a closed three-dimensional manifold where Pesin's entropy formula holds thus establishing the continuous-time version of \cite{T}. Moreover, in any compact manifold of…

We prove that the geodesic flow on a locally CAT(-1) metric space which is compact, or more generally convex cocompact with non-elementary fundamental group, can be coded by a suspension flow over an irreducible shift of finite type with…

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa--Holm equations are well-studied examples.A beautiful approach to well-posedness is…

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is…

We prove for $C^\infty$ non-singular flows on three-dimensional compact manifolds with positive entropy, there are at most finitely many ergodic measures of maximal entropy. This result extends the notable work of Buzzi-Crovisier-Sarig…