Ergodic Theory Research Articles

On the Rate of Convergence of Ergodic Averages for Functions of Gordin Space

For an automorphisms with non-zero Kolmogorov-Sinai entropy, a new class of $L_2$-functions called the Gordin space is considered. This space is the linear span of Gordin classes constructed by some automorphism-invariant filtration of $\sigma$-algebras~$\mathfrak{F}_n$. A function from the Gordin class is an orthogonal projection with respect to the operator $I-E(\cdot|\mathfrak{F}_n)$ of some $\mathfrak{F}_m$-measurable function. After Gordin's work on the use of the martingale method to prove the central limit theorem, this construction was developed in the works of Voln\'{y}. In this review article we consider this construction in ergodic theory. It is shown that the rate of convergence of ergodic averages in the $L_2$ norm for functions from the Gordin space is simply calculated and is $\mathcal{O}(\frac{1}{\sqrt{n}}).$ It is also shown that the Gordin space is a dense set of the first Baire category in ${L_2(\Omega,\mathfrak{F},\mu)\ominus L_2(\Omega,\Pi(T,\mathfrak{F}),\mu)},$ where $\Pi(T,\mathfrak{F})$ is the Pinsker $\sigma$-algebra.

On the ergodic theory of impulsive semiflows

In this work, we established a simpler criterion for the existence of invariant measures for impulsive semiflows. In addition, we exhibit sufficient conditions to obtain a Variational Principle and the existence and uniqueness of equilibrium states. Some examples are provided to illustrate the theory developed.

Characterizations of Measurability-Preserving Ergodic Transformations

Characterizations of Measurability-Preserving Ergodic Transformations

Convergence analysis of the time-varying discrete-time Altafini model on signed network

This paper mainly studies the convergence of time-varying discrete-time Altafini model. Firstly, we convert the bipartite consensus of the Altafini model with n nodes into consensus of the DeGroot model with 2 n nodes by adopting lifting approach. Secondly, based on structural balance theory and node relabelling theory, the consensus of enlarged DeGroot model can be further equivalently converted into the consensus of one simple DeGroot model with n nodes. Thirdly, by using the equivalent transformation and the ergodic coefficient theory, we change the sufficient condition from ‘repeatedly jointly strongly connected’ graphs to ‘repeatedly jointly scrambling’ graphs, and verify that the new condition also can lead to the bipartite consensus of the Altafini model. This novel criterion allows for determining the bipartite consensus of the Altafini model without the condition of ‘repeatedly jointly strongly connected’ graphs. Next, we provide a concise proof for the case of opinions converging to 0 with the condition of ‘repeatedly jointly strongly connected’ by the ergodic coefficient theory. Finally, we develop a reasonable coevolution model to illustrate the effectiveness and strong applicability of our main theoretical results.

Weak ergodicity breaking transition in a randomly constrained model

Experiments in Rydberg atoms have recently found unusually slow decay from a small number of special initial states. We investigate the robustness of such long-lived states (LLS) by studying an ensemble of locally constrained random systems with tunable range μ. Upon varying μ, we find a transition between thermal and weakly nonergodic (supporting a finite number of LLS) phases. Furthermore, we demonstrate that the LLS observed in the experiments disappear upon the addition of small perturbations so that the transition reported here is distinct from known ones. We then show that the LLS dynamics explores only part of the accessible Hilbert space, thus corresponding to localization in Hilbert space. Published by the American Physical Society 2024

Notes on noncommutative ergodic theorems

Given a semifinite von Neumann algebra M \mathcal M equipped with a faithful normal semifinite trace τ \tau , we prove that the spaces L 0 ( M , τ ) L^0(\mathcal M,\tau ) and R τ \mathcal R_\tau are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in L 0 ( M , τ ) L^0(\mathcal M,\tau ) . Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L 1 ( M , τ ) L^1(\mathcal M,\tau ) can be extended to pointwise convergence of such nets in any fully symmetric space E ⊂ R τ E\subset \mathcal R_\tau , in particular, in any space L p ( M , τ ) L^p(\mathcal M,\tau ) , 1 ≤ p > ∞ 1\leq p>\infty . Some applications of these results in the noncommutative ergodic theory are discussed.

Dowker’s ergodic theorem by the Chacon–Ornstein theorem

Dowker’s ergodic theorem by the Chacon–Ornstein theorem

Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets

We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.

Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces

In this paper, we introduce inductive limits of the Fréchet spaces ℓ(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell (p+)$$\\end{document}, ces(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {ces}(p+)$$\\end{document}, and d(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(p+)$$\\end{document} (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p < \\infty $$\\end{document}) and projective limits of the (LB)-spaces ℓ(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell (p-)$$\\end{document}, ces(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {ces}(p-)$$\\end{document}, and d(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(p-)$$\\end{document} (1<p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < p \\le \\infty $$\\end{document}). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} (0≤t≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le t \\le 1$$\\end{document}) act continuously in these sequence spaces, and we determine the spectra. Finally, we study the ergodic properties, that is, power boundedness, (uniform) mean ergodicity, and supercyclicity, of the operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} acting in the (LF)-spaces and in the (PLB)-spaces mentioned above.

Anti-classification results for weakly mixing diffeomorphisms

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the 2-torus.

Modified least squares estimators for Ornstein–Uhlenbeck processes from low-frequency observations

We propose a modified least squares estimator for the drift parameter of the Ornstein–Uhlenbeck process when the observations are available at a discrete instant in a low-frequency level. Unlike in the past literature, this modified least squares estimator is asymptotically unbiased. This estimator is combined with the ergodic theorem to obtain joint estimators (θˆn,σˆn2) for both drift and diffusion parameters (θ,σ2). For any fixed observation time step h>0, the strong consistency and joint asymptotic normality of our estimators (θˆn,σˆn2) are obtained by using the linear model technology and the Delta method. Surprisingly, this linear model technology is not new but it is used for the first time to our model. It very significantly simplifies the arguments that researchers used in the literature. Numerical results illustrate the asymptotic behavior of the estimators.

Multidimensional polynomial patterns over finite fields: Bounds, counting estimates and Gowers norm control

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument.

A digital Technology–Cultural resource strategy to drive innovation in cultural industries: A dynamic analysis based on machine learning

With the gradual maturing of digital technology, the innovation strategy of ‘technology + culture’ is becoming increasingly significant for the sustainable development of cultural industries. This paper takes cultural industries in China's 31 provinces and municipalities in 2019 as a research sample and seeks to explore the multi-dimensional and periodical evolution relationship between digital technology, cultural resources, and innovation in cultural industries by drawing on machine learning algorithms and ergodic theory. The following conclusions are obtained. Firstly, the relationship between digital technology and cultural resources and their impact on the innovation of cultural industries is found to be nonlinear, which is a dynamic evolution process. Secondly, the impact of cultural resources on cultural industry innovation is not completely positively correlated, and the impact on technological innovation is more like an inverted U-shaped curve with a rising, peak, and declining period; the impact on content innovation, meanwhile, is similar to the ferment and take-off stages in the S-curve of innovation. And digital technology plays a significant role in promoting innovation, and is, especially in technological innovation, more like an ascending step from quantitative changes to qualitative change. Thirdly, digital technology and cultural resources are found to play different roles in the different stages of innovative development. Cultural resources are the basic factor and digital technology is the incentive factor.

A mathematical introduction to complex dynamical systems

This work proposes an investigation of the theory of dynamic systems, focusing on the analysis of entropy in its classical and topological manifestations. Initially, the fundamental concepts of dynamical systems theory are presented, with an emphasis on topological dynamical systems (TDS). Next, definitions of discrete topological entropy and its relationship to dynamical systems are discussed, followed by the introduction of topological entropy pressure as a weighted form of topological entropy. Subsequently, some applications of topological entropy in dynamic systems are examined, highlighting its role in the analysis of chaotic systems and in ergodic theory. Furthermore, we present a new theory, called Topological Ergodic Entropy Theory (TEET), which offers an innovative approach to the analysis of ergodic dynamical systems. Finally, the Ergodic Theory of Turbulent Flow (ETTF) is introduced, exploring the relationship between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. The results presented contribute to a deeper understanding of the complexity of dynamical systems and their applications in various areas of mathematics and physics. The investigation of topological entropy and its applications in dynamical systems offers new insights into the chaotic and stochastic behavior of these systems, while the introduction of new theories, such as ETTF, provides a new perspective for the analysis and modeling of turbulent phenomena.

A statistical learning framework for mapping indirect measurements of ergodic systems to emergent properties

The discovery of novel experimental techniques often lags behind contemporary theoretical understanding. In particular, it can be difficult to establish appropriate measurement protocols without analytic descriptions of the underlying system-of-interest. Here we propose a statistical learning framework that avoids the need for such descriptions for ergodic systems. We validate this framework by using Monte Carlo simulation and deep neural networks to learn a mapping between nuclear magnetic resonance spectra acquired on a novel low-field instrument and proton exchange rates in ethanol-water mixtures. We found that trained networks exhibited normalized-root-mean-square errors of less than 1 % for exchange rates under 150 s−1 but performed poorly for rates above this range. This differential performance occurred because low-field measurements are indistinguishable from one another for fast exchange. Nonetheless, where a discoverable relationship between indirect measurements and emergent dynamics exists, we demonstrate the possibility of approximating it in an efficient, data-driven manner.

New expectation-based conditions for almost sure exponential stability of discrete-time semi-Markovian linear systems with time-delayed control

This paper studies almost sure exponential stability of discrete-time semi-Markovian linear systems with time-delayed control. New expectation-based sufficient conditions for almost sure exponential stability of the system are derived by employing the coupled Lyapunov function related to the sojourn time and the ergodic property of semi-Markov chains. Compared with known results on almost sure exponential stability of discrete-time semi-Markovian linear systems, the results in this paper are shown to be less conservative. At last, three numerical examples are given to illustrate the results.

Decay of time correlations in point vortex systems

The dynamics of a large point vortex system whose initial configuration consists in uniformly distributed independent positions is investigated. Time correlations of local observables of the vortex configuration are shown to be compatible with power law decay 1/t, providing additional insight on ergodicity and mixing properties of equilibrium dynamics in point vortex models.

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy–Wolff theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, we address such questions in the very general setting of sequences (Fn) of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. One of our main results is new even in the classical setting.Our methods apply in particular to non-autonomous dynamical systems, when (Fn) are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics.The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a ‘weak independence’ version of the second Borel–Cantelli lemma and the concept from ergodic theory of ‘shrinking targets’.

Additive and geometric transversality of fractal sets in the integers

AbstractBy juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of ‐invariant sets on the 1‐torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include: a classification of all subsets of the positive integers that are simultaneously ‐ and ‐invariant; integer analogs of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection and the sumset of ‐ and ‐invariant sets and when and are multiplicatively independent; and a description of the dimension of iterated sumsets for any ‐invariant set . We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the ‐dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.

Immersion in Complex Dynamical Systems with Ergodicity

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.