Paper finds positive metric entropy in perturbed geodesic flow.
problem Understanding dynamics outside KAM tori in nearly integrable systems.
method Lagrangian perturbation of geodesic flow on a flat 3-torus.
result Positive metric entropy found outside some KAM tori.
The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under C0 perturbations. method Study of topological entropy on Riemannian metrics with C0 topology. result Metrics with contractible closed geodesics have robust entropy.
New Finsler flow on 2-torus has chaotic dynamics.
problem Constructing chaotic dynamics on a 2-torus.
method Using Berger and Turaev's theorem, constructing a Finsler metric.
result Found a Finsler geodesic flow with positive metric entropy.
Study on flexibility of metric entropy and geodesic flow constraints on surfaces.
problem Flexibility and constraints of metrics and flows on surfaces.
method Analysis of geodesic flow, entropy, and metrics in a fixed conformal class.
result Additional restrictions on topological entropy and new geometric properties.
Note proves bi-Lipschitz embeddings for Ham(S^2) metrics.
problem Entropy norms on Ham(S^2) bi-Lipschitz embeddings.
method Proves bi-Lipschitz embeddings for Ham(S^2) with entropy metrics.
result Existence of bi-Lipschitz embeddings for Ham(S^2) metrics.
An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by C∞ functions and has positive topological entropy is constructed.
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…
Study K3 surfaces and their metrics, focusing on dynamics.
problem Understanding dynamics on K3 surfaces.
method Interactions between K3 surface geometry and Ricci-flat metrics, dynamical study of automorphisms.
result Positive entropy automorphisms on K3 surfaces.
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. 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…
The paper studies minimal surface entropy on hyperbolic 3-manifolds and compares it to the hyperbolic case.
problem Minimal surface entropy on hyperbolic 3-manifolds and its comparison to the hyperbolic case.
method Analysis of Ricci flow convergence and comparison of metrics with sectional and scalar curvature constraints.
result The entropy is maximized at the hyperbolic metric under certain curvature conditions.
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
The paper constructs quasimorphisms on surface diffeomorphism groups with entropy bounds.
problem Entropy bounds on surface diffeomorphism groups.
method Constructing homogeneous quasimorphisms and defining entropy metrics.
result There are infinitely many linearly independent quasimorphisms with entropy bounds.
The paper shows dense and residual sets of continuous maps with positive metric mean dimension.
problem Understanding the genericity of continuous maps with positive metric mean dimension.
method Analyzing continuous maps on compact Riemannian manifolds and Cantor sets.
result The set of continuous maps with metric mean dimension equal to a given value is dense and, for the dimension, residual in the space of continuous maps.
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…
In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 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…
We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.
Researchers derive expressions for metric perturbations of extremal surfaces.
problem Understanding changes in extremal surfaces under metric perturbations.
method Derived explicit expressions for position and surface area changes.
result Found an expansion of surface area involving multiple integrals of geometric quantities.
Static spacetimes are stable attractors in a flow equation.
problem Stability of static spacetimes with negative cosmological constant.
method New expander entropy for Ricci-harmonic flow.
result Static metrics are stable if and only if a positive mass theorem holds.
Liouville entropy increases strictly along Ricci flow on surfaces.
problem Understanding the behavior of Liouville entropy under Ricci flow.
method New expression for Liouville entropy derivative, proof of positivity in specific directions.
result Liouville entropy is strictly increasing along normalized Ricci flow for 1/6-pinched metrics.
The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.
problem Modeling chaotic positional dynamics of stars in celestial systems.
method Discrete dynamical systems, Ricci flow, Perelman entropy, Lyapunov exponents, bifurcation analysis.
result Entropy increases exponentially, indicating challenging long-term star position prediction.
This paper defines the pressure metric on the Moduli space of Margulis spacetimes without cusps and shows that it is positive definite on the constant entropy sections. It also demonstrates an identity regarding the variation of the cross-ratios.
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 C1+α diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nod…
The paper constructs and proves the nondecreasing property of functionals for conformal Ricci flow.
problem Nondecreasing property of functionals for conformal Ricci flow.
method Constructed two functionals for positive solutions to the conjugate heat equation. Proved nondecreasing property by calculating explicit evolution formulas and establishing pointwise formulas.
result Nondecreasing property of functionals for conformal Ricci flow, with strict increase only for Einstein metrics.
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.
problem Estimating gradients of solutions to a specific type of partial differential equation.
method Using the Finslerian Allen-Cahn equation as an Euler-Lagrange equation to a Liapunov entropy functional, proving gradient estimates on compact and noncompact Finsler metric measure spaces.
result Global and local gradient estimates of positive solutions to the Finslerian Allen-Cahn equation.
Positive entropy automorphisms have virtual eigenvalues outside the unit circle.
problem Understanding the dynamics of surface automorphisms with positive entropy.
method Investigating virtual homological eigenvalues of surface automorphisms.
result Existence of virtual eigenvalues outside the unit circle for automorphisms with positive entropy.
Most simple braids have positive topological entropy.
problem Understanding the topological entropy of simple braids.
method Reduction from simple braids to non-simple 3-strand braids.
result The proportion of simple braids with positive entropy approaches 100% as the number of strands increases.
We consider magnetic flows on 2-step nilmanifolds M=Γ\G, where the Riemannian metric g and the magnetic field σ are left-invariant. Our first result is that when σ represents a rational cohomology class and its restriction to g=TeG vanishes on the derived algebra, then the associated…
Entropy norm on surface diffeomorphisms is unbounded.
problem Bounding entropy norm on surface diffeomorphisms.
method Analyzing the group of diffeomorphisms of a closed orientable surface.
result Entropy norm is unbounded on the group of diffeomorphisms of a closed orientable surface.
The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.
problem Conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds on non-smooth spaces.
method Description of results in dimension 3, exploration of weak forms of Ricci curvature, use of volume entropy and Bishop-Gromov inequality.
result Recent results on weak Ricci curvature bounds and conditions for positive or non-negative scalar curvature in 3-manifolds.
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic f…
The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.
problem Investigating strong positive recurrence in negatively curved manifolds.
method Defining and comparing entropy and pressure at infinity through different measures.
result Strong positive recurrence potentials admit finite Gibbs measures.
We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent…
Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.
problem Integrability and entropy compactness for Kähler potentials with specific density.
method Skoda-Zeriahi type integrability theorem and log-log threshold detection.
result Positivity of integrability threshold and entropy compactness for uniform log-log threshold.
Explicit bounds on group size for certain geometric actions.
problem Bounding group size in geometric actions with bounded entropy.
method Proving an explicit function F(k, E) for groups with k-acylindrical splittings.
result Groups with bounded entropy have a finite size bound.
Paper improves Frank-Wolfe algorithm's efficiency bounds.
problem Establishing efficient iteration complexity for Frank-Wolfe algorithm.
method Using metric entropy to provide lower bounds.
result Frank-Wolfe requires many iterations for certain problems.
The study examines the flexibility of entropies for negatively curved surfaces.
problem The study investigates the flexibility of topological and metric entropies for negatively curved surfaces.
method The authors compare different metrics on surfaces of negative curvature and analyze their topological and metric entropies.
result The study proves that the topological and metric entropies for metrics of negative curvature are flexible and only equal in the case of constant negative curvature.
Metric graphs have subgraphs with entropy at least λ.
problem Finding subgraphs with high entropy in metric graphs.
method Proving existence of subgraphs with entropy at least λ for graphs of rank r with entropy 1.
result Metric graphs have subgraphs with entropy at least λ.
Paper constructs new non-Anosov Partially Hyperbolic Geodesic flows using conformal deformations.
problem Creating new non-Anosov Partially Hyperbolic Geodesic flows.
method Using conformal deformations to produce examples of partially hyperbolic geodesic flows.
result Proves ergodicity for the Liouville measure and uniqueness of the measure of maximal entropy.
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. Paper characterizes embeddability of function spaces into Lp-type RKBS via metric entropy.
problem Characterizing embeddability of function spaces into Lp-type RKBS. method Establishes a connection between metric entropy growth and embeddability.
result A bound on metric entropy growth allows embedding into Lp-type RKBS. This paper characterizes mu-cscK metrics using Perelman's W-entropy.
problem Characterizing mu-cscK metrics and understanding their properties.
method Using Perelman's W-entropy as a functional on the tangent bundle of Kähler metrics, the paper characterizes mu-cscK metrics as critical points of this functional.
result The W-entropy is monotonic along geodesics and provides a lower bound for mu-entropy.
We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for C1+ε flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact C∞ surface has at least const ×(ehT/T) simple clos…
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.
problem Understanding entropy changes in flows near hyperbolic metrics.
method Analysis of geodesic flow on Riemannian manifolds with variable negative curvature.
result Topological entropy strictly decreases along normalized Ricci flow near hyperbolic metrics.
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.