Entropies of non-positively curved metric spaces

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 packing condition. Similar estimates will be given in case of closed subsets of the boundary of Gromov-hyperbolic metric spaces with convex geodesic bicombings

Entropies of non positively curved metric spaces

We show the equivalences of several notions of entropy, such as 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 packing condition. Similar estimates will be given in case of closed subsets of the boundary of Gromov-hyperbolic metric spaces with convex geodesic bicombings. A uniform Ahlfors regularity of the limit set of quasiconvex-cocompact actions on Gromov-hyperbolic packed metric spaces with convex geodesic bicombing will be shown, implying a uniform rate of convergence to the entropy. As a consequence we prove the continuity of the critical exponent for quasiconvex-cocompact groups with bounded codiameter

GH-convergence of CAT(0)(0)-spaces: stability of the Euclidean factor

We prove that if a sequence of geodesically complete CAT$(0)$-spaces $X_j$ with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $X_\infty$, then the dimension of the maximal Euclidean factor splitted off by $X_\infty$ and $X_j$ is the same, for $j$ big enough. In other words, no additional Euclidean factors can appear in the limit.Comment: arXiv admin note: text overlap with arXiv:2307.0564

Ahlfors regular conformal dimension and Gromov-Hausdorff convergence

We prove that the Ahlfors regular conformal dimension is upper semicontinuous with respect to Gromov-Hausdorff convergence when restricted to the class of uniformly perfect, uniformly quasi-selfsimilar metric spaces. A corollary is the upper semicontinuity of the Ahlfors regular conformal dimension of limit sets of discrete, quasiconvex-cocompact group of isometries of uniformly bounded codiameter of $\delta$-hyperbolic metric spaces under equivariant pointed Gromov-Hausdorff convergence of the spaces.Comment: Minor revision

Packing conditions in metric spaces with curvature bounded above and applications

General metric spaces satisfying weak and synthetic notions of upper and lower curvature bounds will be studied. The relations between upper and lower bounds will be pointed out, especially the interactions between a packing condition and different forms of convexity of the metric. The main tools will be a new and flexible definition of entropy on metric spaces and a version of the Tits Alternative for groups of isometries of the metric spaces under consideration. The applications can be divided into classical and new results: the former consist in generalizations to a wider context of the theory of negatively curved Riemannian manifolds, while the latter include several compactness and continuity results

A geometric approach to Poincar\'e inequality and Minkowski content of separating sets

The goal of this paper is to continue the study of the relation between the Poincar\'e inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincar\'e inequality and energy of separating sets, arXiv 2401.02762]. A new shorter proof is provided. An intermediate tool is the study of the lower bound of another geometric quantity, called separating ratio. The main novelty is the description of the relation between the infima of the separating ratio and the Minkowski content of separating sets. We prove a quantitative comparison between the two infima in the local quasigeodesic case and equality in the local geodesic one. No Poincar\'e assumption is needed to prove it. The main tool employed in the proof is a new function, called the position function, which allows in a certain sense to fibrate a set in boundaries of separating sets. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the p-Poincar\'e inequality, by proving along the way equivalence with a notion of p-pencil that extends naturally the definition for p = 1.Comment: 38 pages, 4 figure

Convergence and collapsing of CAT(0)(0)-group actions

We study the theory of convergence for groups $\Gamma$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces $X$, and for their quotients $M=\Gamma \backslash X$ (CAT$(0)$-orbispaces). We describe splitting and collapsing phenomena for nonsingular actions, explaining how they occurs and when the limit action is discrete. This leads to finiteness results for nonsingular actions on uniformly packed CAT$(0)$-spaces with uniformly bounded codiameter, which generalize and sharpen, in nonpositive curvature, the classical finiteness theorems of Riemannian geometry. Finally, we prove some closure and compactness theorems in the class of CAT$(0)$-homology orbifolds, and an isolation result for flats

Poincar\'{e} inequality and energy of separating sets

We study geometric characterizations of the Poincar\'{e} inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the $1$-Poincar\'e inequality.Comment: 37 pages, 1 figure. Comments are welcom

The Metric Completion of the Space of Vector-Valued One-Forms

The space of full-ranked one-forms on a smooth, orientable, compact manifold (possibly with boundary) is metrically incomplete with respect to the induced geodesic distance of the generalized Ebin metric. We show a distance equality between the induced geodesic distances of the generalized Ebin metric on the space of full-ranked one-forms and the corresponding Riemannian metric defined on each fiber. Using this result we immediately have a concrete description of the metric completion of the space of full-ranked one-forms. Additionally, we study the relationship between the space of full-ranked one-forms and the space of all Riemannian metrics, leading to quotient structures for the space of Riemannian metrics and its completion

Volume entropy and rigidity of locally symmetric Riemannian manifolds with negative curvature

This thesis is based on the article "Entropie minimale et rigiditès des espaces localement symétriques de courbure strictement négative" of G.Besson, G.Courtois and S.Gallot. We consider a compact oriented n-Rimennian manifold M that supports a locally symmetric metric of negative sectional curvature. The aim of the thesis is to show that an invariant characterizes this metric. The invariant that we consider is the volume entropy of a metric g on a compact Riemannian manifold. It measures the growth of the volume of the balls B(x,r) of the covering manifold of M endowed with the cover metric when r tends to infinity. The entropy of a metric g is denoted by h(g). The quantity ent(g)=h(g)^n*Vol(M,g), where Vol(M,g) is the volume of M calculated with respect to g, is invariant under homotheties. The main result of the thesis is that the locally symmetric metric of negative curvature on M minimizes the quantity ent(g), where g varies among all the Riemannian metrics on M. Moreover it is unique in the following sense: if g is a Riemannian metric that minimizes the functional ent, then, up to homotheties, the metric g is isometric to the locally symmetric one. This result give us some rigidity theorems on Riemannian manifolds that support a locally symmetric metric of negative curvature. For example a corollary of the theorem above is the well known Mostow rigidity theorem