The paper studies geodesics on surfaces with unique foliations, proving limit cycles.
problem Understanding geodesics on surfaces with unique foliations.
method Conditions for infinite quasi-geodesics, limit representations, and measure projections.
result Existence of limit cycles in the Thurston boundary.
The paper examines limit sets of Weil-Petersson geodesics and their properties.
problem Understanding the limit sets of Weil-Petersson geodesics.
method Proving properties of limit sets and constructing examples.
result The limit set of a Weil-Petersson geodesic ray is a single point, except for minimal nonuniquely ergodic cases.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to R-trees and estimate their dimensions. result Estimates the dimensions of maximal simplices for both rational and irrational trees.
The abstract discusses nonuniqueness results for specific Riemannian invariants.
problem Identifying conditions for nonhomothetic conformal rescalings with constant Riemannian invariants.
method Identifying sufficient conditions for finite and infinite geometrically distinct periodic conformal rescalings.
result Improves and establishes nonuniqueness results for various Riemannian invariants.
Lectures on mean curvature flow and its related equations.
problem Singularity formation, nonuniqueness, and topological change in motion by mean curvature.
method Analyzes motion by mean curvature flow and related equations.
result Exploration of singularity formation, nonuniqueness, and topological change.
Paper solves dual Minkowski problem in 2D plane for specific curvature cases.
problem Finding the number of solutions to the dual Minkowski problem in 2D with constant curvature.
method Combining theoretical analysis and numerical estimation of an integral with parameters.
result Found the number of solutions for the constant dual curvature case when 0<q≤4. In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe metrics and with TT-tensor σ = 0 have a non-trivial solution, and thus answer a que…
New findings show non-uniqueness in Ricci flow solutions for dimensions n≥5.
problem Non-uniqueness in Ricci flow solutions for dimensions n≥5.
method Exhibited a family of asymptotically conical gradient shrinking solitons with non-unique forward continuations.
result Non-uniqueness of Ricci flow solutions for dimensions n≥5.
We extend the notion of what it means for a complete Ricci flow to have a given initial metric, and consider the resulting well-posedness issues that arise in the 2D case. On one hand we construct examples of nonuniqueness by showing that surfaces with cusps can evolve either by keeping the cusps or by contracting them…
The paper explores when end summing is unique for manifolds at infinity.
problem Nonuniqueness of end summing in manifolds with specific fundamental group behavior at infinity.
method Examines various categories (TOP, PL, DIFF) and types of 0- and 1-handles.
result Uniqueness is proved for Mittag-Leffler ends and generalized to handle slides and cancellations.
Researchers found multiple ways to end-sum 4-manifolds, contradicting a previous conjecture.
problem Nonuniqueness of end-sums in 4-manifolds.
method Explicit examples and detailed discussion of end-cohomology algebra.
result Uncountably many distinct proper homotopy types from end-sums.
We give existence and nonuniqueness results for simple planar curves with prescribed geodesic curvature.
This paper classifies solutions to a specific hyperbolic geometry problem.
problem Classifying solutions to a specific hyperbolic geometry equation.
method Analytical and numerical methods to solve the equation.
result Classification of solutions for p≥−7, nonuniqueness for p<−7. Study finds multiple conformal metrics with constant Q-curvature.
problem Finding complete conformal metrics with constant Q-curvature.
method Analyzing manifolds of dimension ≥5, including spheres and complex projective spaces.
result Infinitely many branches of metrics with constant Q-curvature, bifurcating from Berger metrics.
Researchers solve Yamabe problems for specific operators, finding both uniqueness and nonuniqueness.
problem Prescribing scalar, Q-, or σ₂-curvatures in conformal classes.
method Formally self-adjoint, conformally covariant, polydifferential operators.
result Uniqueness results on the sphere, nonuniqueness in general.
The paper confirms conjectures about ancient ovals and provides counterexamples.
problem Understanding the uniqueness and nonuniqueness of ancient ovals under different symmetries.
method Analyzing mean curvature flow solutions and constructing symmetric ancient ovals.
result Confirms conjectures about ancient ovals and provides counterexamples.
Because of the relevance of the results, this paper is merged into the paper titled "On the Number of Solutions to Asymptotic Plateau Problem" (arXiv:math.DG/0505593) as a new section.
Riemann moduli spaces are quantum ergodic for certain dimensions.
problem Quantum ergodicity of Riemann moduli spaces.
method Analysis of Weil--Petersson metric and geodesic flow.
result Riemann moduli spaces Mg,n are quantum ergodic for 3g+n≥4. Study counts ergodic measures in surface lamination strata.
problem Counting ergodic measures in surface lamination strata.
method Determined through analysis of geodesic laminations.
result Number of ergodic measures identified in each stratum.
Example shows unique ergodicity of foliations on surfaces.
problem Unique ergodicity of foliations on surfaces.
method Construction of example and geometric criterion.
result Horizontal foliation is uniquely ergodic under certain conditions.
Logarithmic laws affect ergodicity in Teichmüller geodesics.
problem Ergodicity of translation flows in Teichmüller spaces.
method Analysis of flat systole vs. geodesic length.
result Flat geometry ensures unique ergodicity, unlike hyperbolic geometry.
Recent results on ergodic theory for Riemann surface laminations and foliations.
problem Ergodic theorems for laminations and foliations on Riemann surfaces.
method Leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles, Lyapunov exponents.
result Definition and study of canonical Lyapunov exponents for singular holomorphic foliations.
New progress on frame flow ergodicity for nearly pinched manifolds.
problem Ergodicity of frame flow on negatively-curved manifolds.
method New ideas leading to ergodicity for nearly 0.25-pinched manifolds.
result Achieved progress towards Brin's conjecture.
Study finds non-uniqueness in sphere metrics with constant fractional curvature.
problem Non-uniqueness of metrics with constant positive fractional curvature on spheres.
method Bifurcation techniques applied to non-local equations with critical non-linearity.
result Non-uniqueness results for complete metrics on Sn∖Sk. Asymptotically consistent clustering algorithms for ergodic stochastic processes are developed.
problem Clustering stochastic processes with consistency guarantees.
method Review and development of clustering algorithms for ergodic stochastic processes.
result Asymptotically consistent clustering algorithms can be obtained for ergodic stochastic processes.
Formula connects foliated simplicial volume with group cost.
problem Calculating integral foliated simplicial volume.
method Ergodic decomposition formula for simplicial volume.
result Integration formula linking foliated simplicial volume and group cost.
We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quan…
Strong stability of ergodic iterations proven without ergodic driving sequence.
problem Ensuring strong stability of ergodic iterations under non-ergodic driving sequences.
method Revisiting processes driven by stationary ergodic sequences, proving strong stability under mild conditions on recursive maps.
result Strong stability of iterations proven without ergodic driving sequence.
The paper studies the ergodicity of frame flow on even-dimensional manifolds.
problem Understanding the ergodicity of frame flow on even-dimensional manifolds.
method Analyzing pinching conditions to determine ergodicity.
result The frame flow is ergodic under specific pinching conditions for even-dimensional manifolds.
Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
Percolation study in non-hyperbolic groups proves non-uniqueness phase.
problem Percolation in acylindrically hyperbolic groups.
method Analyzing Bernoulli bond percolation on Cayley graphs of groups.
result Non-uniqueness phase in percolation on Cayley graphs of acylindrically hyperbolic groups.
Log-ergodic model improves velocity of money prediction.
problem Improving velocity of money prediction for economic control.
method Log-ergodic processes to simulate monetary velocity.
result Log-ergodic model offers superior predictive power.
'Ergodicity economics' is criticized as pseudoscience.
problem Flawed conceptual basis of mainstream economic theory.
method Claims 'ergodicity economics' is more parsimonious and clearer.
result Peters' approach has not produced falsifiable implications.
New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.
problem Stabilizing quantum ergodicity for complex systems.
method Combines mixed quantization techniques with stable ergodicity results for partially hyperbolic systems.
result Establishes stable quantum ergodicity for spin Hamiltonians.
The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to S1. result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.
We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the foldin…
The paper studies ergodicity of flows on subspaces, generalizing earlier work.
problem Ergodicity of flows on subspaces of higher rank groups.
method Analyzes one-parameter diagonalizable subgroups of connected semisimple groups acting on homogeneous spaces.
result Obtains an ergodicity criterion similar to Hopf-Tsuji-Sullivan for general Anosov subgroups.
Quantum ergodicity study improves Lp norms of eigenfunction restrictions.
problem Improving Lp norms of eigenfunction restrictions on submanifolds. method Local Lp estimates and improvements for quantum ergodic eigenfunctions. result Logarithmic improvements on negatively curved manifolds, o(1) on ergodic geodesic flows. A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this ar…
Establishes conditions for geometric ergodicity in Hamiltonian Monte Carlo.
problem Conditions for geometric ergodicity in Hamiltonian Monte Carlo.
method Analyzes Markov chains with position-independent and position-dependent integration times.
result Recovery of geometric ergodicity in Hamiltonian Monte Carlo for broader tail behaviors.
A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (u…
Method quantifies spectral ergodicity in deep learning networks.
problem Understanding the success of deep learning architectures.
method Combines TM and KL divergence metrics to analyze random matrix ensembles.
result Spectral ergodicity increases with network size, suggesting its importance.
The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.
problem Function theory on Teichmüller space and dynamics of mapping class groups.
method Utilizes Thurston's theory and Sullivan's theory on discrete subgroups of hyperbolic space.
result Establishes connections between function theory, dynamics, and ergodic theory.
The geodesic flow on certain surfaces is shown to be ergodic.
problem Ergodicity of geodesic flows on surfaces without focal points.
method Analyzing geodesic flows on surfaces with no focal points, proving ergodicity under specific curvature conditions.
result The geodesic flow on the unit tangent bundle of a surface with no focal points is ergodic with respect to the Liouville measure.
The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.
problem Stable ergodicity of group actions on smooth manifolds restricted to one-dimensional cases.
method Geometric method using quasi-conformal blender for constructing stable local dynamics.
result Every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class C1+α. This work investigates a mixture of LMC and RMHMC with MMALA for geometric ergodicity.
problem Lack of geometric ergodicity study in Riemannian manifold and Lagrangian Monte Carlo methods.
method Investigates a mixture of LMC and RMHMC with MMALA to achieve geometric ergodicity.
result Demonstrates geometric ergodicity in the mixture of LMC and RMHMC with MMALA.
Study finds multiple periodic solutions to ODEs related to curvature problems.
problem Finding multiple positive periodic solutions to quasilinear ODEs.
method Global bifurcation techniques applied to second order quasilinear ODEs.
result Bifurcation-theoretic proof of nonuniqueness for conformal metrics with constant scalar curvature.
Study shows non-wandering, partially hyperbolic systems are ergodic.
problem Ergodicity of partially hyperbolic systems.
method Analysis of partially hyperbolic diffeomorphisms, focusing on non-wandering systems.
result These systems are ergodic when they preserve volume, confirming a conjecture.