Explains Lyapunov exponents and their importance in three fields.
problem None explicitly stated, focuses on explaining concepts.
method Expository writing based on talks and research by Artur Avila.
result Lyapunov exponents are crucial in smooth ergodic theory, Teichmüller theory, and spectral theory.
Proves compact Cauchy horizons have constant surface gravity under null energy condition.
problem Proving compact Cauchy horizons have constant surface gravity.
method Combines ergodic theory, Hodge theory, and Riemannian flow theory.
result Compact Cauchy horizons admit a smooth lightlike tangent vector field of constant surface gravity.
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.
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…
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.
We introduce the concept of solenoid as an abstract laminated space. We do a thorough study of solenoids, leading to the notion of ergodic and uniquely ergodic solenoids. We define generalized currents associated with immersions of oriented solenoids with a transversal measure into smooth manifolds, generalizing Ruelle…
Let {Tt} 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 {Tt} is Bernoulli, or {Tt} is isomorphic to the product of a Bernoulli flow and a rotational flow. Appli…
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…
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. 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. The paper provides a link between ergodic theory and symplectic topology. A classical notion of ergodic theory is a skew product map associated with a loop in a group of transformations. We study skew products which come from loops in the group of Hamiltonian diffeomorphisms of a symplectic manifold. Our main question …
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.
Quantum theory explains price dynamics in financial markets, capturing bid-ask spread and ergodicity.
problem Nature of price formation in financial markets and bid-ask spread dynamics.
method Developed a quantum coupled-wave theory using a 2x2 price operator with eigenvalues representing bid and ask prices.
result The theory adequately models bid-ask spread and directional price movement due to quantum-chaotic interaction.
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.
We extend Schwartzman theory beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions of solenoids with a trapping region.
New framework for improved inference algorithms.
problem Challenges in improving VI and MCMC methods.
method Ergodic Inference framework based on ergodic transformations.
result Established theoretical foundation for practical algorithms.
'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.
Smoothly conjugate perturbations of certain toral automorphisms.
problem Smoothly conjugating perturbations of partially hyperbolic toral diffeomorphisms.
method Study of perturbations of partially hyperbolic toral automorphisms with smooth center foliations.
result Existence of smooth leaf conjugacy for small perturbations of L with smooth center foliation.
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.
The paper tackles learning optimal predictions from a single trajectory of a stochastic dynamical system.
problem Learning from a single finite trajectory of an ergodic stochastic dynamical system.
method The approach involves estimating the optimal one-step prediction function using nonlinear least squares and deriving high-probability guarantees.
result The study provides high-probability guarantees for the optimal prediction function, accounting for the non-independent and non-identically distributed nature of trajectory data.
Extends earthquake and horocycle flows to new measures.
problem Ergodic theory of earthquake flow on measured laminations.
method Generalizes shear coordinates to arbitrary measured laminations.
result Classifies ergodic measures for P action on bundle of quadratic differentials.
New test compares data to ergodic Markov models without specifying an alternative.
problem Testing goodness of fit for ergodic Markov processes without an alternative model.
method Density-based test comparing data to specified models' stationary densities.
result Test provides new insights into econometric and financial modeling.
This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of requiring a piecewise smooth path over which to integrate, we can now do so over …
3-manifolds' volumes match stable integral values.
problem Determining 3-manifold volumes accurately.
method Integral foliated simplicial volume and ergodic theory.
result 3-manifolds' volumes equal stable integral simplicial volumes.
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
The paper tackles learning to control systems with unknown parameters using Brownian noise.
problem Learning to control systems with unknown parameters.
method Proposes algorithms based on moving empirical averages and integrates statistical methods with stochastic control theory.
result Achieves a logarithmic expected regret rate.
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.
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …
Investigates spontaneous symmetry breaking in non-equilibrium systems.
problem Spontaneous symmetry breaking of ergodicity in non-equilibrium systems.
method Mathematical and effective field theory approaches to investigate symmetry breaking.
result Symmetry breaking phenomena observed in stochastic processes.
Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.
We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinite-dimensional vector space of link invariants. In con…
We develop the intersection theory associated to immersed, oriented and mea- sured solenoids, which were introduced in arXiv:0910.2836.
Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.
problem Ergodicity and mixing properties of Anosov flows and their isometric extensions.
method Microlocal analysis of Pollicott-Ruelle resonances to study isometric extensions of Anosov flows.
result Ergodicity of frame flow on negatively-curved Riemannian manifolds under specific curvature assumptions.
Study of flows on circle bundles over translation surfaces, showing decay of correlations.
problem Ergodic properties of flows on circle bundles over translation surfaces.
method Generalizing Heisenberg nilflows to more general base surfaces, showing relatively mixing.
result Showed that such flows exhibit decay of correlations in the orthogonal complement of functions constant along fibers.
Last iterate of Extragradient algorithm converges slower than averaged iterates in saddle point problems.
problem Smooth convex-concave saddle point problems
method Analysis of Extragradient (EG) algorithm convergence rates
result The last iterate of EG converges at a rate of O(1/√T), compared to O(1/T) for averaged iterates
The Manhattan curve connects metrics of hyperbolic groups, showing rigidity.
problem Understanding the relationship between different metrics on hyperbolic groups.
method Ergodic theory of topological flows and analysis of Patterson-Sullivan measures.
result The Manhattan curve is a straight line if and only if metrics are roughly similar.
The paper establishes CLTs for Markov chains and improves sampling algorithms for heavy-tailed distributions.
problem Establishing central limit theorems for ergodic averages of Markov chains.
method Drift conditions to provide necessary and sufficient conditions for CLTs, including lower bounds on convergence rates.
result Sharp conditions and convergence rates for various MCMC algorithms on heavy-tailed targets.
We consider the smoothing probabilities of hidden Markov model (HMM). We show that under fairly general conditions for HMM, the exponential forgetting still holds, and the smoothing probabilities can be well approximated with the ones of double sided HMM. This makes it possible to use ergodic theorems. As an applicatio…
Extends magnetic flow theory results to higher dimensions.
problem Magnetic flows on manifolds of arbitrary dimension.
method New Pestov identities and adapted Riemannian geometry concepts.
result Established tensor tomography and ergodicity results for magnetic flows.
Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate finite sets of points at a predefined resolution gamma > 0. We establish a connection between the g…
Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we …
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
Study approximates top Lyapunov exponents for surface mapping classes.
problem Approximating topological Lyapunov exponents for surface mapping classes.
method Periodic approximation and joint spectral radius extension.
result Top Lyapunov exponents can be approximated by periodic orbits.
The paper studies invariant measures for specific actions in algebraic groups.
problem Investigating invariant measures for horospherical actions and Anosov groups.
method Analyzing the space of invariant measures for NM-actions on Γ\G. result The space of invariant measures is homeomorphic to RextrankG−1. New algorithm learns optimal resource allocation in wireless systems without models.
problem Learning optimal resource allocation in wireless systems without system models.
method Developed a model-free primal-dual algorithm using smoothed surrogates of constrained problems.
result The algorithm can make the gap between optimal values and dual values arbitrarily small.
Develops stability conditions for estimating affine jump-diffusions.
problem Ergodicity and consistency of parameter estimation for affine jump-diffusions.
method Establishes stochastic stability conditions and ergodicity under specific conditions.
result Proves strong laws of large numbers and functional central limit theorems for additive functionals.
Study uniform learnability of binary classification networks with communication.
problem Learning a network with communication between vertices from uniform ergodic Random Graph Process.
method Introduced structural Rademacher complexity and used martingale method and Marton's coupling.
result Uniform learnability as worst-case theoretical limits for binary classification problems.
The paper solves investment problems with uncertain factors using game theory.
problem Optimal forward investment in an incomplete market with model uncertainty.
method Combining stochastic differential games and ergodic BSDE approach.
result Representation of robust forward performance processes in factor form.