ergodic analysis

This study investigates the preservation of several classical combinatorial properties—such as recurrence, morphicity, and factor frequencies—under the action of deterministic finite-state transducers on infinite words. To address this problem, the authors introduce a unified analytical framework by combining Krohn–Rhodes decomposition theory with ergodic methods from symbolic dynamics for the first time. Within this framework, they systematically characterize the capacity of transducers to preserve these combinatorial properties and establish a comprehensive set of preservation theorems encompassing all the aforementioned features. This work provides a theoretical foundation for understanding the structural stability of sequences under automaton-based transformations.

本文通过量化替代过程中的信息量并使用合适的并行计算模型，提供了一种判断多维符号动力系统是否为sofic系统的有效方法。

Classical probabilistic constructions—such as almost-invariant σ-algebras, ergodic decompositions, the de Finetti theorem, and the zero–one law—lack a unified structural explanation within standard probability theory. Method: We construct a category whose objects are probability spaces and whose morphisms are measure-preserving Markov kernels, identified up to almost-sure equality. Our approach integrates categorical methods (notably the Markov category and its dagger structure), standard Borel space theory, and null-isomorphism techniques. Contribution/Results: We provide the first structural, categorical proof of the ergodic decomposition theorem; characterize almost-invariant σ-algebras uniformly as both limits and colimits; establish a dual limit–colimit characterization of invariant structures in random dynamical systems; and unify three foundational limit theorems—the de Finetti theorem, the zero–one law, and the ergodic decomposition—within a single abstract categorical framework. This advances the structural coherence, universality, and intrinsic categorical nature of probability theory.

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.

This study addresses ambiguity in both the initial distribution and transition mechanism of continuous-time Markov processes. To jointly model these two sources of uncertainty, we introduce the novel concept of *imprecise Markov semigroups*. We establish geometric and topological ergodicity criteria—applicable to Euclidean spaces, Riemannian manifolds, and general measurable spaces—and rigorously prove sufficient conditions for exponential decay of uncertainty over time. Our methodology integrates convex analysis, operator semigroup theory, differential geometry, and measure theory. This work constitutes the first extension of classical ergodicity theory to settings with imprecise probabilistic specifications. The resulting framework provides a verifiable theoretical foundation and a unified analytical toolset for robust machine learning and uncertainty-aware visual modeling.

This study investigates the existence and well-definedness of asymptotic frequencies of arbitrary finite patterns in smooth sequences over the alphabet {1,3}. By introducing a type classification based on local structure, the authors construct a substitution system corresponding to the associated subshift and combine tools from ergodic theory and symbolic dynamics to establish criteria for minimality and unique ergodicity. The main contribution is the proof that the asymptotic frequency of every finite pattern in {1,3}-smooth sequences always exists and is uniquely determined by the derived type sequence, thereby providing a rigorous mathematical characterization of the statistical regularities inherent in smooth sequences.

This study investigates the computability of the maximal ergodic average and the set of maximizing measures in zero-temperature ergodic optimization. Under the assumption of a computable potential function and reasonable regularity conditions, it is proven that the maximal ergodic average is a computable real number and that the set of maximizing measures forms a Π₁-computable compact set. Focusing on finite-range interactions over subshifts of finite type, the work establishes the first computability framework for this setting and provides an explicit algorithm capable of computing these quantities exactly within finite time. Accompanying the theoretical results, executable code implementing the algorithm is also made available.

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.

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 a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.