Related papers: Volume entropy semi-norm

We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for $C^1$ flows, every sectional hyperbolic set $\Lambda$ is entropy expansive, and the topological entropy varies continuously with the…

In arxiv:1205.1274 Rieck and Yamashita defined the link volume of 3-manifolds and studied some of its basic properties. Many of these properties are similar to the corresponding properties of the hyperbolic volume. In this paper we…

Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially…

Entropy of all systems that we understand well is proportional to their volumes except for black holes given by their horizon area. This makes the microstates of any quantum theory of gravity drastically different from the ordinary matter.…

In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $n$-space can have. We prove some useful equalities and inequalities on this volume. For the regular…

We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described.…

We establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of Sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples.

Given a $k$-self similar set $X\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.

A theorem of Gutt-Hutchings-Ramos asserts that all normalized symplectic capacities give the same value for monotone four-dimensional toric domains. We generalize this theorem to arbitrary dimension. The new ingredient in our proof is the…

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $\Sigma$ of a Riemannian 5-manifold $M$…

We introduce the notion of tubular dimension, and give a formula for it. As an application we show that every invariant measure of a $C^{1+\gamma}$ diffeomorphism of a closed Riemannian manifold admits an asymptotic local product structure…

Topological entropy is a widely studied indicator of chaos in topological dynamics. Here we give a generalized definition of topological entropy which may be applied to set-valued functions. We demonstrate that some of the well-known…

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

We study the embedding $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided that $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. Based on this finding, we…

We prove a complete realization theorem for multifractal entropy spectra of continuous potentials on a broad class of dynamical systems. More precisely, for every $H>0$ and every continuous concave function on a compact interval with…

We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of…

After reformulating Gromov's non-squeezing theorem as an area-inequality, we discuss a seemingly natural higher dimensional generalization.

We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along…

We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds…

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov…