ergodic theory

This study addresses the challenge that multiplicative weights update algorithms in games often fail to converge to Nash equilibria and exhibit unpredictable long-term behavior due to Li-Yorke chaos. To overcome this, we introduce for the first time the natural invariant measure from ergodic theory into the analysis of game dynamics. This framework not only characterizes strategy frequencies but also precisely computes long-run time averages of economically relevant observables—such as payoffs, social cost, and regret—even in the absence of pointwise convergence. Focusing on two-strategy congestion games, we rigorously establish that the system retains statistical predictability and provide a unified description of its full dynamical spectrum, ranging from periodic attractors to coexisting chaotic regimes, thereby revealing the algorithm’s capacity to replicate canonical behaviors of one-dimensional dynamical systems.

This work addresses the lack of statistical fidelity in time-series modeling of dynamical systems, where conventional generalization error fails to guarantee accurate reproduction of physical invariants—such as invariant measures and Lyapunov exponents. To this end, we establish, for the first time, an ergodic-theoretic statistical generalization framework, explicitly identifying preservation of the physical measure as the central learning objective. We uncover a fundamental cause of statistical distortion in Neural ODEs applied to dynamical system regression: their neglect of Jacobian structural constraints. We theoretically prove—and empirically validate—that Jacobian regularization substantially enhances statistical generalization. On benchmark chaotic systems (Lorenz, Rössler), incorporating Jacobian information reduces estimation errors in invariant measures, statistical moments, and Lyapunov exponents by an average of 47%, outperforming MLPs, ResNets, Fourier Neural Networks, and RNNs.

Invariant measures in the original state coordinates fail to uniquely identify chaotic dynamical systems due to topological conjugacy ambiguity, as they only capture asymptotic statistical behavior. Method: Leveraging Takens’ embedding theorem, we reconstruct invariant measures in delay-coordinate space, thereby enhancing their sensitivity to underlying dynamics. Contribution/Results: We first prove that the invariant measure constructed from a single scalar time series in delay coordinates suffices for system identification up to topological conjugacy. Furthermore, by jointly constructing delay-coordinate invariant measures from multiple observables, we eliminate conjugacy ambiguity entirely, achieving unique identification. This approach integrates ergodic theory with constructive measure design, markedly improving discriminative power—especially under realistic constraints such as observational noise, sparse sampling, and initial-condition uncertainty. It extends both the theoretical foundations and practical applicability of measure-theoretic methods in dynamical systems modeling.

This study investigates the fundamental cause of uniqueness of invariant probability measures for Markov kernels on general state spaces. Employing a purely measure-theoretic approach, it introduces “indecomposability” as the core structural property ensuring uniqueness, reinterpreting classical irreducibility as a sufficient but not necessary condition. By leveraging the mutual singularity among distinct ergodic invariant measures and their common absolute continuity with respect to a reference measure, the work establishes a uniqueness criterion in standard Borel spaces that dispenses with assumptions of recurrence, regeneration, or kernel regularity. This framework both simplifies and extends classical ergodic theory, eliminating reliance on traditional dynamical systems tools.

This work addresses the problem of unsupervised learning of low-dimensional, manipulable dynamical system representations—namely, compact and smooth state variables coupled with differentiable vector fields—directly from raw video, without prior physical knowledge or domain-specific assumptions. We propose the first end-to-end, video-driven framework for manipulable dynamics discovery, integrating neural implicit state modeling, contrastive spatiotemporal regularization, and differential-geometric constraints to jointly ensure state interpretability, dynamical differentiability, and behavioral analyzability. Evaluated across diverse dynamical systems—including chaotic, limit-cycle, stable fixed-point, and natural oscillatory regimes—the method accurately recovers essential dynamical features (e.g., attractors, bifurcations, conserved quantities) and achieves significantly higher long-horizon prediction accuracy than existing baselines.

This work addresses the challenge of modeling dynamical systems subject to state-dependent, non-i.i.d., and non-Gaussian noise by proposing a general identification framework. By integrating dynamical system embedding theory with random feature mappings, the method extends classical noise-free system identification approaches to complex stochastic environments. It establishes that only \(2p+1\) random features are sufficient to uniquely identify continuous or discrete-time dynamical models containing \(p\) parameters. Theoretical analysis provides identifiability guarantees for a broad class of stochastic dynamical systems, while numerical experiments on the Lorenz-63 system and Hénon map demonstrate the method’s efficacy in accurately recovering underlying system structures from observations corrupted by strongly correlated, non-Gaussian noise.

This work addresses the statistical learning challenges posed by non-i.i.d. data arising from a single finite trajectory of an ergodic stochastic dynamical system, focusing on one-step-ahead prediction modeling. By employing nonlinear least squares to estimate the predictive function and leveraging the invariant measure and uniform geometric ergodicity of the underlying Markov process, the study establishes the first high-probability generalization error bound for non-i.i.d. trajectories grounded in the system’s invariant measure. The approach integrates concentration inequalities for Hilbert space-valued additive functionals with Koopman operator approximation. The theoretical guarantees apply broadly across settings including higher-order systems and finite state spaces, thereby significantly extending the applicability of statistical learning theory to dynamical systems.

Chaotic systems are highly sensitive to modeling errors, making it challenging to simultaneously achieve accuracy in local dynamics and long-term statistical fidelity. To address this, this work proposes a novel approach that constructs a local cover of the chaotic attractor in phase space and jointly optimizes both the Jacobian accuracy of the surrogate model and its long-term statistical behavior. The method uniquely integrates local pushforward distribution matching with Jacobian fidelity, thereby unifying local and global modeling paradigms. Training employs a loss function based on distribution matching, utilizing the Maximum Mean Discrepancy (MMD) metric. Experimental results demonstrate that the proposed method substantially improves Jacobian accuracy while achieving state-of-the-art performance in long-term statistical properties.

This work addresses the fundamental limits of symbolic discovery of dynamical system governing equations, demonstrating that these limits are governed by the geometric structure of attractors rather than solely by algorithmic choices or data volume. The study proposes the smallest eigenvalue, λ_min(M), of the moment matrix of the invariant measure as a universal identifiability bound, revealing for the first time that attractor geometry fundamentally constrains equation discovery through this quantity. Leveraging the Birkhoff ergodic theorem to compute λ_min(M), the authors validate its algorithm-agnostic nature using SINDy and PySR on Lorenz-84 and Lorenz-96 systems, and introduce a Soft F1-weighted structural scoring metric to discern performance differences invisible to conventional metrics. Results show that while chaos enhances λ_min(M), noise sensitivity varies across algorithms, and the proposed framework enables cross-system transferability without retraining.

This work proposes a novel approach based on Takens’ phase space reconstruction to characterize the multifractality and scale invariance of stochastic processes. By constructing ensembles of neighboring analog states around target states, the method jointly analyzes the volume-based probability distribution and Lagrangian dispersion dynamics, thereby establishing—for the first time—a direct link between the geometric structure of phase space and the statistical properties of the underlying process. The framework is successfully applied to fractional Brownian motion and multifractal random walks, accurately recovering their scaling exponents and demonstrating that the geometry of the reconstructed phase space is fundamentally governed by the process’s multifractal characteristics. This provides a new perspective for characterizing the dynamics of complex stochastic systems.