arXiv research

Most simple braids have positive topological entropy.

Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.

The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.

An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^\infty$ functions and has positive topological entropy is constructed.

We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+ε}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{hT}/T)$ simple clos…

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

The study examines conditions for minimal volume entropy of simplicial complexes.

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic…

The paper examines robustness of topological entropy in geodesic flows.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

We discuss a class of (local and non-local) theories of gravity that share same properties: i) they admit the Einstein spacetime with arbitrary cosmological constant as a solution; ii) the on-shell action of such a theory vanishes and iii) any (cosmological or black hole) horizon in the Einstein spacetime with a positi…

New Finsler flow on 2-torus has chaotic dynamics.

The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.

We prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface of positive genus is unbounded.

In this note we prove that for each positive integer $m$ there exists a bi-Lipschitz embedding $Z^m\to Ham(S^2)$, where $Ham(S^2)$ is equipped with the entropy metric. In particular, the same result holds when the entropy metric is substituted with the autonomous metric.

Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Appli…

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a th…

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces.

We consider the entropy of the solution to the heat equation on a Riemannian manifold. When the manifold is compact, we provide two estimates on the rate of change of the entropy in terms of the lower bound on the Ricci curvature and the spectral gap respectively. Our explicit computation for the three dimensional hype…

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

Information theory provides principled ways to analyze different inference and learning problems such as hypothesis testing, clustering, dimensionality reduction, classification, among others. However, the use of information theoretic quantities as test statistics, that is, as quantities obtained from empirical data, p…

This review explores entropy applications in data analysis and machine learning.

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…

The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.

The study bounds entanglement entropy for coherent states on Kähler manifolds.

This study uses moving average cluster entropy to analyze financial market dynamics.

News novelty predicts negative stock market returns.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

Study K3 surfaces and their metrics, focusing on dynamics.

Symplectic homology matches dual capacities for convex domains.

The abstract applies waist inequality to dynamical systems and entropy.

In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.

New algorithm estimates semi-continuous data density using entropy maximization.

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed…

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.

Unified framework for analyzing gradient flows of measures with exponential decay of entropy.

Liouville entropy increases strictly along Ricci flow on surfaces.

The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

New method uses SVD entropy to price artworks.

A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+α}$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nod…

Pointwise localization allows more precise localization and accurate interpretability, compared to bounding box, in applications where objects are highly unstructured such as in medical domain. In this work, we focus on weakly supervised localization (WSL) where a model is trained to classify an image and localize regi…

Static spacetimes are stable attractors in a flow equation.

New method calibrates reference distributions for bounded support.

Survey on Ricci flow on spaces with conical singularities.