Ergodic Theory Research Articles

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.

On the failure of Ornstein theory in the finitary category

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [Finitary isomorphism of m m -dependent processes, Amer. Math. Soc., Birkhauser, Providence, RI, 1992, pp. 373–376], which was first suggested by D. Rudolph [A characterization of those processes finitarily isomorphic to a Bernoulli shift, Birkhäuser, Boston, Mass., 1981, pp. 1–64]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai’s factor theorem for ergodic processes. Another consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [Math. Inst. Hautes Études Sci. 128 (2018), 1–119].

Ergodic criterion of a random diffusivity model.

The random diffusivity, initially proposed to explain Brownian yet non-Gaussian diffusion, has garnered significant attention due to its capacity not only for elucidating the internal physical mechanism of non-Gaussian diffusion, but also for establishing an analytical framework to characterize particle motion in complex environments. In this paper, based on the correlation function C(t_{1},t_{2})=〈D(t_{1})D(t_{2})〉 of random diffusivity D(t), we quantitatively propose a general criterion of determining the ergodic property of the Langevin equationwith the arbitrary random diffusivity D(t). Due to the critical role of correlation function C(t_{1},t_{2}), we derive the criterion for the two cases with stationary diffusivity or nonstationary diffusivity, respectively. By utilizing the quantitative criterion, we can directly judge the ergodic properties of the random diffusivity model based on the correlation function C(t_{1},t_{2}) of random diffusivity D(t). Several typical diffusivities, including the common square of the Brownian motion and of the (fractional) Ornstein-Uhlenbeck process, are found to contribute to different ergodic properties, which validates our proposed criterion built on the correlation function C(t_{1},t_{2}).

A class of kth‐order dependence‐driven random coefficient mixed thinning integer‐valued autoregressive process to analyse epileptic seizure data and COVID‐19 data

SummaryData related to the counting of elements of variable character are frequently encountered in time series studies. This paper brings forward a new class of th‐order dependence‐driven random coefficient mixed thinning integer‐valued autoregressive time series model (DDRCMTINAR()) to deal with such data. Stationarity and ergodicity properties of the proposed model are derived in detail. The unknown parameters are estimated by conditional least squares, and modified quasi‐likelihood and asymptotic normality of the obtained parameter estimators is established. The performances of the adopted estimate methods are checked via simulations, which present that modified quasi‐likelihood estimators perform better than the conditional least squares considering the proportion of within‐ estimates in certain regions of the parameter space. The validity and practical utility of the model are investigated by epileptic seizure data and COVID‐19 data of suspected cases in China.

Some notes on ergodic theorem for [formula omitted]-statistics of order [formula omitted] for stationary and not necessarily ergodic sequences

In this note, we give sufficient conditions for the almost sure and the convergence in Lp of a U-statistic of order m built on a strictly stationary but not necessarily ergodic sequence.

The Covariance Metric in the Blaschke Locus

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichmüller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in Guillarmou et al. in (Ergod Theory Dyn Syst 43:974–1022, 2021). We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.

Arcsine Laws of Light.

We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy's arcsine laws-a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).PPRHAI2643-1564]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.

Convergence of Langevin-simulated annealing algorithms with multiplicative noise

We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : R d → R V : \mathbb {R}^d \to \mathbb {R} a potential function to minimize, we consider the stochastic differential equation d Y t = − σ σ ⊤ ∇ V ( Y t ) dY_t = - \sigma \sigma ^\top \nabla V(Y_t) d t + a ( t ) σ ( Y t ) d W t + a ( t ) 2 Υ ( Y t ) d t dt + a(t)\sigma (Y_t)dW_t + a(t)^2\Upsilon (Y_t)dt , where ( W t ) (W_t) is a Brownian motion, where σ : R d → M d ( R ) \sigma : \mathbb {R}^d \to \mathcal {M}_d(\mathbb {R}) is an adaptive (multiplicative) noise, where a : R + → R + a : \mathbb {R}^+ \to \mathbb {R}^+ is a function decreasing to 0 0 and where Υ \Upsilon is a correction term. This setting can be applied to optimization problems arising in Machine Learning; allowing σ \sigma to depend on the position brings faster convergence in comparison with the classical Langevin equation d Y t = − ∇ V ( Y t ) d t + σ d W t dY_t = -\nabla V(Y_t)dt + \sigma dW_t . The case where σ \sigma is a constant matrix has been extensively studied; however little attention has been paid to the general case. We prove the convergence for the L 1 L^1 -Wasserstein distance of Y t Y_t and of the associated Euler scheme Y ¯ t \bar {Y}_t to some measure ν ⋆ \nu ^\star which is supported by argmin ⁡ ( V ) \operatorname {argmin}(V) and give rates of convergence to the instantaneous Gibbs measure ν a ( t ) \nu _{a(t)} of density ∝ exp ⁡ ( − 2 V ( x ) / a ( t ) 2 ) \propto \exp (-2V(x)/a(t)^2) . To do so, we first consider the case where a a is a piecewise constant function. We find again the classical schedule a ( t ) = A log − 1 / 2 ⁡ ( t ) a(t) = A\log ^{-1/2}(t) . We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.

Decision Support System Driven by Thermo-Complexity: Scenario Analysis and Data Visualization

The present modelling aims to construct a computational information representation system useful for decision support system (DSS) solutions in the realization of intelligent systems or complex systems analysis solutions. Starting from an n-dimensional space (with n ≥ 7) represented by problem variables (referred to as CSF—Critical Success Factors), a dimensional embedding procedure is used to transition to a two-dimensional space. In the two-dimensional space, thanks to new lattice motion algorithms, the decision support system can determine the optimal solution with a lower computational cost based on the decision-maker’s preferences. Finally, thanks to an algorithm that takes into account the hierarchical order of importance of the seven CSFs as per the expert’s liking or according to his optimization logics, a return is made to the n-dimensional space and the final solution in the original space. As we will see, the starting and ending states in the n-dimensional space (referred to as micro-states) when projected into the two-dimensional space generate states (referred to as macro-states) which are degenerate. In other words, the correspondence between micro-states and macro-states is not one-to-one, as multiple micro-states correspond to one macro-state. Therefore, in relation to the decision-maker’s preferences, it will be the responsibility of the decision support system to provide the decision-maker with the micro-state of interest in the n-dimensional space (dimensional emergence procedure), starting from the obtained optimal macro-state. This result can be achieved starting from a flat chain of sensors capable of measuring/emulating certain specific parameters of interest. As we will see, it emerges that by considering random–exhaustive rolling value paths in order to track and potentially intervene to rebalance a dynamic system representing the state of stress/sensing of a system of interest, we are using the most general and, therefore, complex hypotheses of ergodic theory. In this work, we will focus on the representation of information in n-dimensional and two-dimensional spaces, as well as construct evaluation scenarios. We will also show the results of the decision support system in some cases of specific interest, thanks to a specific lattice motion algorithm of the realized decision-making environment.

Decay of long-lived oscillations after quantum quenches in gapped interacting quantum systems

The presence of long-lived oscillations in the expectation values of local observables after quantum quenches has recently attracted considerable attention in relation to weak ergodicity breaking. Here, we focus on an alternative mechanism that gives rise to such oscillations in a class of systems that support kinematically protected gapped excitations at zero temperature. An open question in this context is whether such oscillations will ultimately decay. We provide strong support for the decay hypothesis by considering spin models that can be mapped to systems of weakly interacting fermions, which in turn are amenable to an analysis by standard methods based on the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. We find that there is a time scale beyond which the oscillations start to decay that grows as the strength of the quench is made small. Published by the American Physical Society 2024

Ergodic properties of multiplication and weighted composition operators on spaces of holomorphic functions

AbstractWe obtain different results about the mean ergodicity of weighted composition operators when acting on the spaces , , and , where is the open unit ball of a Banach space, as well as about the compactness and the mean ergodicity of the multiplication operator. This study relates these properties of the operators with properties of the symbol and the weight defining such operators.

Diffusion of active Brownian particles under quenched disorder.

The motion of a single active particle in one dimension with quenched disorder under the external force is investigated. Within the tailored parameter range, anomalous diffusion that displays weak ergodicity breaking is observed, i.e., non-ergodic subdiffusion and non-ergodic superdiffusion. This non-ergodic anomalous diffusion is analyzed through the time-dependent probability distributions of the particle's velocities and positions. Its origin is attributed to the relative weights of the locked state (predominant in the subdiffusion state) and running state (predominant in the superdiffusion state). These results may contribute to understanding the dynamical behavior of self-propelled particles in nature and the extraordinary response of nonlinear dynamics to the externally biased force.

Random invariant densities for markov operator cocycles and random mean ergodic theorem

In this paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.

On stationary random processes with fuzzy states

In this paper, continuous random processes with fuzzy states are studied. The properties of their numerical characteristics – fuzzy expectations, expected values and covariance functions – are established. The main attention is paid to the class of stationary fuzzy-random processes. For them, the ergodicity property and the spectral representation of covariance function (generalized Wiener–Khinchin theorem) are substantiated. The results obtained are based on the properties of fuzzy-random variables and numerical random processes. Triangular fuzzy-random processes are considered as examples.

Linear progress in fibres

A fibred hyperbolic 3-manifold induces a map from the hyperbolic plane to hyperbolic 3-space, the respective universal covers of the fibre, and the manifold. The induced map is an embedding that is exponentially distorted in terms of the individual metrics. In this article, we begin a study of the distortion along typical rays in the fibre. We verify that a typical ray in the hyperbolic plane makes linear progress in the ambient metric in hyperbolic 3-space. We formulate the proof in terms of some soft aspects of the geometry and basic ergodic theory. This enables us to extend the result to analogous contexts that correspond to certain extensions of closed surface groups. These include surface group extensions that are Gromov hyperbolic, the universal curve over a Teichmüller disc, and the extension induced by the Birman exact sequence.

Strong Ergodicity in Nonhomogeneous Markov Systems with Chronological Order

In the present, we study the problem of strong ergodicity in nonhomogeneous Markov systems. In the first basic theorem, we relax the fundamental assumption present in all studies of asymptotic behavior. That is, the assumption that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix. In addition, we study the practically important problem of the rate of convergence to strong ergodicity for a nonhomogeneous Markov system (NHMS). In a second basic theorem, we provide conditions under which the rate of convergence to strong ergodicity is geometric. With these conditions, we in fact relax the basic assumption present in all previous studies, that is, that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix geometrically fast. Finally, we provide an illustrative application from the area of manpower planning.

An undecidable statement regarding zero-sum games

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist “paradoxical” two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory.

The evolving butterfly: Statistics in a changing attractor

The Earth system is often modeled as a dynamical system in what has come to be known as Earth System Models. When used to study anthropogenically forced climate change, these models are forced in such a way that they are not in a statistically stationary state. Yet, statistical statements are still made about the Earth climate system using only a single trajectory by taking temporal averages. At each moment in time, one draws a sample from a different distribution, raising questions about the utility of temporal averages, in stark contrast to the utility of temporal averages in ergodic systems. This work follows in the tradition of using a toy model to examine properties present in the Earth climate system. We aim to examine how we can make meaningful statistical statements in non-stationary systems when only dealing with a single trajectory. We use the Lorenz equations with a time-varying parameter as a starting point for comparing ensemble averages to temporal averages. We find that, in so far as the control parameter induces a slow and smooth change in the dynamics, the resulting statistics of ensemble averages compare well to those of temporal averages.

Some Generalized Entropy Ergodic Theorems for Nonhomogeneous Hidden Markov Models

Entropy measures the randomness or uncertainty of a stochastic process, and the entropy rate refers to the limit of the time average of entropy. The generalized entropy rate in the form of delayed averages can overcome the redundancy of initial information while ensuring stationarity. Therefore, it has better practical value. A Hidden Markov Model (HMM) contains two stochastic processes, a stochastic process in which all states can be observed and a Markov chain in which all states cannot be observed. The entropy rate is an important characteristic of HMMs. The transition matrix of a homogeneous HMM is unique, while a Nonhomogeneous Hidden Markov Model (NHMM) requires the transition matrices to be dependent on time variables. From the perspective of model structure, NHMMs are novel extensions of homogeneous HMMs. In this paper, the concepts of the generalized entropy rate and NHMMs are defined and fully explained, a strong limit theorem and limit properties of a norm are presented, and then generalized entropy ergodic theorems with an almost surely convergence for NHMMs are obtained. These results provide concise formulas for the computation and estimation of the generalized entropy rate for NHMMs.