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The violation of detailed balance poses a serious problem for the majority of current quasiclassical methods for simulating nonadiabatic dynamics. In order to analyze the severity of the problem, we predict the long-time limits of the electronic populations according to various quasiclassical mapping approaches by applying arguments from classical ergodic theory. Our analysis confirms that regions of the mapping space that correspond to negative populations, which most mapping approaches introduce in order to go beyond the Ehrenfest approximation, pose the most serious issue for reproducing the correct thermalization behavior. This is because inverted potentials, which arise from negative electronic populations entering the nuclear force, can result in trajectories unphysically accelerating off to infinity. The recently developed mapping approach to surface hopping (MASH) provides a simple way of avoiding inverted potentials while retaining an accurate description of the dynamics. We prove that MASH, unlike any other quasiclassical approach, is guaranteed to describe the exact thermalization behavior of all quantum–classical systems, confirming it as one of the most promising methods for simulating nonadiabatic dynamics in real condensed-phase systems. [ABSTRACT FROM AUTHOR]

We explore the existence of spin glass phase in FeCo/FePt bilayers arising due to disordered ferromagnet. The non-ergodic and highly degenerate landscape of the spin glass phase at low temperature explains the origin of complex magnetic texture in the FeCo/FePt system. Upon cooling the bilayered system, the magnetic texture undergoes spin freezing below 120 K as evident from the bifurcations in zero field cooling and field cooling magnetizations at low magnetic field as a manifestation of broken ergodicity. The uncompensated magnetic moments originating in the spin glass state result in slow time dynamics of thermoremanent magnetization. Consequently, the bilayers demonstrate an intriguing magnetic memory effect in which the magnetic state of the system could be retrieved upon isothermal ageing below 120 K after reversing the temperature cycle. Thermal treatment deteriorates the spin glass behaviour and shows a transition to strong ferromagnetic character in FeCo/FePt bilayers. [ABSTRACT FROM AUTHOR]

In this paper we propose some Harris-like criteria in order to study the long time behavior of general positive and periodic semiflows. These criteria allow us to obtain new existence results of principal eigenelements, and their exponential attractiveness. We present applications to two biological models in a space-time varying environment: a non local selection-mutation equation and a growth-fragmentation equation. The particularity of this article is to study some inhomogeneous problems that are periodic in time, as it appears for instance when the environment changes, due for instance to the seasonal cycle or circadian rhythms. [ABSTRACT FROM AUTHOR]

This letter raises the possibility that ergodicity concerns might have some bearing on the signal‐to‐noise paradox. This is explored by applying the ergodic theorem to the theory behind ensemble weather forecasting and the ensemble mean. Using the ensemble mean as our best forecast of observations amounts to interpreting it as the most likely phase‐space trajectory, which relies on the ergodic theorem. This can fail for ensemble forecasting systems if members are not perfectly exchangeable with each other, the averaging window is too short and/or there are too few members. We argue these failures can occur in cases such as the winter North Atlantic Oscillation (NAO) forecasts due to intransitivity or regime behaviour for regions such as the North Atlantic and Arctic. This behaviour, where different ensemble members may become stuck in different relatively persistent flow states (intransitivity) or multi‐modality (regime behaviour), can in certain situations break the ergodic theorem. The problem of non‐ergodic systems and models in the case of weather forecasting is discussed, as are potential mitigation methods and metrics for ergodicity in ensemble systems. [ABSTRACT FROM AUTHOR]

A family I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal I on X is below an ideal J on Y in the Katětov order if there is a function f : Y → X such that f - 1 [ A ] ∈ J for every A ∈ I . We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent. [ABSTRACT FROM AUTHOR]

From a dynamical systems point of view, a trivalent spin network model in Loop Quantum Gravity is considered, which presents self‐organized criticality (SOC), arising from a spin propagation dynamics. A partition function is obtained for the domains of stability connecting gauge non‐invariant avalanches, leading to an entropy formula for the asymptotic SOC state. The microscopic origin of this SOC entropy is therefore given by the excitation‐relaxation spin dynamics in the avalanche cycle. The puncturing of trivalent spin networks (TSN) edges participating in the avalanche are counted in terms of an ensemble perimeter over the implicit avalanches. By identifying this perimeter with that of an isolated (2+1)$(2+1)$‐D black hole horizon, it is conjectured that the SOC entropy reduces to the Bekenstein‐Hawking perimeter‐entropy law for the Bañados, Teitelboim, and Zanelli (BTZ) black hole, by an appropriate adjustment of a potential function based on the thermodynamical formalism of Sinai, Ruelle, and Bowen. [ABSTRACT FROM AUTHOR]

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$ , $n\in {\mathbb {Z}}$ , $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the specification property for $ (\alpha,\beta ) $ (α,β)-shifts. When $ \alpha =0 $ α=0, Schmeling shows that the set of $ \beta \gt 1 $ β>1 for which the β-shift has the specification property has the Lebesgue measure zero but has the full Hausdorff dimension [Schmeling. Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory Dyn. Syst. 17 (1997), pp. 675–694]. So it is natural to ask what happens when $ \alpha \gt 0 $ α>0. Buzzi shows that for fixed α the set of $ \beta \gt 1 $ β>1 for which the $ (\alpha,\beta ) $ (α,β)-shift has the specification property has Lebesgue measure zero. Hence we consider the Hausdorff dimension of the parameter space of $ (\alpha,\beta ) $ (α,β)-shifts. [ABSTRACT FROM AUTHOR]

In this paper, we study the mean Li-Yorke chaotic phenomenon along any infinite positive integer sequence for infinite-dimensional random dynamical systems. To be precise, we prove that if an injective continuous infinite-dimensional random dynamical system (X , ϕ) over an invertible ergodic Polish system (Ω , F , P , θ) admits a ϕ -invariant random compact set K with positive topological entropy, then given a positive integer sequence a = { a i } i ∈ N with lim i → + ∞ ⁡ a i = + ∞ , for P -a.s. ω ∈ Ω there exists an uncountable subset S (ω) ⊂ K (ω) and ϵ (ω) > 0 such that for any two distinct points x 1 , x 2 ∈ S (ω) , one has lim inf N → + ∞ 1 N ∑ i = 1 N d (ϕ (a i , ω) x 1 , ϕ (a i , ω) x 2) = 0 , lim sup N → + ∞ 1 N ∑ i = 1 N d (ϕ (a i , ω) x 1 , ϕ (a i , ω) x 2) > ϵ (ω) , where d is a compatible complete metric on X. To overcome the difficulties arising in the infinite-dimensional random dynamical systems, we use the regularity of measure to construct the expected measurable partitions and to decompose the distance function. [ABSTRACT FROM AUTHOR]

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function f every invariant probability measure that maximizes the space average of f must have zero entropy. We establish the analogical result in the context of constrained ergodic optimization, which is introduced by [Garibaldi and Lopes, Functions for relative maximization, Dyn. Syst. 22(4) (2007), pp. 511-528]. [ABSTRACT FROM AUTHOR]

We discuss different regularities on stable/unstable holonomies of cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism has continuous joint stable/unstable holonomies, then it has a dense set of periodic points in its non-wandering set. For that, we prove that the hyperbolic cw-metric (introduced in Artigue et al (2024 J. Differ. Equ. 378 512–38)) can be adapted to be self-similar (as in Artigue (2018 Ergodic Theory Dyn. Syst. 38 2422–46)) and, in this case, continuous joint stable/unstable holonomies are pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms assuming that the stable/unstable holonomies are isometric. In the case the ambient space is a surface, we prove that a cw F -hyperbolic homeomorphism has continuous joint stable/unstable holonomies when every bi-asymptotic sector is regular. [ABSTRACT FROM AUTHOR]

The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota's results on the author's research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations. [ABSTRACT FROM AUTHOR]

This work investigates the stability properties of Lyapunov exponents of transfer operator cocycles from a measure-theoretic perspective. Our results focus on so-called Blaschke product cocycles, a class of random dynamical systems amenable to rigorous analysis. We show that prevalence of stability is related to the dimension of the base system's domain, $ \Omega $. When $ \Omega = S^1 $, we show that stability is prevalent among smooth monic quadratic Blaschke product cocycles fixing the origin by constructing a so-called probe. For higher dimensional $ \Omega $, we show that a probe does not exist, thus providing strong evidence that stability is not prevalent in this setting. Finally, through a perturbative method we show that almost every smooth Blaschke product cocycle fixing the origin is stable. [ABSTRACT FROM AUTHOR]

We consider a class of interval maps with up to two indifferent fixed points, an unbounded derivative, and regularly varying tails. Under some mild assumptions, we prove the existence of a unique mixing absolutely continuous invariant measure and give conditions under which the measure is finite. Moreover, in the finite measure case, we give a formula for the measure-theoretical entropy and upper bounds for a very slow decay of correlations. This extends former work by Coates, Luzzatto, and Muhammad to maps with regularly varying tails. Particularly, we investigate the boundary case where the behaviour of the slowly varying function decides on the finiteness of the measure and on the decay of correlations. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results. [ABSTRACT FROM AUTHOR]

We develop a theory of the critical point of the ferromagnetic Ising model, whose basic objects are the ergodic (pure) states of the infinite system. It proves the existence of anomalous critical fluctuations, for dimension ν = 2 and, under a standard assumption, for ν = 3 , for the model with nearest-neighbor interaction, in a way which is consistent with the probabilistic approach of Cassandro, Jona-Lasinio, and several others, reviewed in Jona-Lasinio's article in Phys. Rep. 352,439 (2001). We propose to single out the state at the critical temperature T c , among the ergodic thermal states associated to temperatures 0 ≤ T ≤ T c , by a condition of non-summable clustering of the connected two-point function. The analogous condition on the connected (2r)- point functions, for r ≥ 2 , together with a scaling hypothesis, natural within our framework, proves that the (macroscopic) fluctuation state is quasi-free, after a proper rescaling, also at the critical temperature, in agreement with a theorem by Cassandro and Jona-Lasinio, whose proof is, however, shown to be incomplete. Other subjects treated include topics relating to universality, including spontaneous breaking of continuous symmetries and violations of universality in problems of energetic and dynamic stability. [ABSTRACT FROM AUTHOR]

Given a semifinite von Neumann algebra \mathcal M equipped with a faithful normal semifinite trace \tau, we prove that the spaces L^0(\mathcal M,\tau) and \mathcal R_\tau are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in 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(\mathcal M,\tau) can be extended to pointwise convergence of such nets in any fully symmetric space E\subset \mathcal R_\tau, in particular, in any space L^p(\mathcal M,\tau), 1\leq p<\infty. Some applications of these results in the noncommutative ergodic theory are discussed. [ABSTRACT FROM AUTHOR]

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$. In this article, we study the unitary frame flow , that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$ -bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$ , there exists $\unicode{x3bb} (m) \in (0, 1)$ such that if $(M, g)$ has negative $\unicode{x3bb} (m)$ -pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\unicode{x3bb} (m)$ satisfy $\unicode{x3bb} (6) = 0.9330...$ , $\lim _{m \to +\infty } \unicode{x3bb} (m) = {11}/{12} = 0.9166...$ , and $m \mapsto \unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60 (1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension. [ABSTRACT FROM AUTHOR]

In this article, we investigate the nonparametric Nadaraya-Watson estimator for the drift function of stochastic differential equations driven by fractional Brownian motion of the Hurst parameter H ∈ (1 / 4 , 1). The drift function is a one-sided dissipative Lipschitz that ensures the ergodic property for the stochastic differential equation. The explicit formula of the estimator is obtained by using the Wick product based on the discretely observed process, which is of the utmost importance for practical applications. With the proper bandwidth selectors, we derive the strong consistency of the proposed estimator, and the main tools are ergodic theory and Malliavin calculus. [ABSTRACT FROM AUTHOR]

Motivated by questions asked by Erdős, we prove that any set A\subset \mathbb {N} with positive upper density contains, for any k\in \mathbb {N}, a sumset B_1+\cdots +B_k, where B_1, ..., B_k\subset \mathbb {N} are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of k=2. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The given text discusses the Three-Distance Theorem, which states that a sequence of fractional parts of points, together with 1, partitions the unit interval into intervals with at most three different lengths. The text explores the finite words that encode the lengths of these intervals and their relationship to three-interval exchanges. It introduces the concept of symmetric discrete interval exchange and provides examples to illustrate the encoding process. The text also discusses cyclic restrictions in permutations and presents a proof of a theorem. It further discusses the properties of sequences and sub-sequences, as well as the concept of distance encoding and its relationship to sequences of distances. The text concludes by discussing the use of dynamical systems and combinatorial tools in the study of interval exchange transformations. [Extracted from the article]

Let be a countable ergodic group of automorphisms of a measure space and be the normalizer of its full group . Problem: for a pair of measurable partitions and of the space , when does there exist an element such that ? For a wide class of measurable partitions, we give a solution to this problem in the case where is an approximately finite group with finite invariant measure. As a consequence, we obtain results concerning the conjugacy of the commutative subalgebras that correspond to and in the type factor constructed via the orbit partition of the group . [ABSTRACT FROM AUTHOR]

The Bucaramanga seismic nest has the highest concentration in volume of intermediate depth seismic events with light and moderate magnitudes (Mb > 4.8) worldwide. Despite the nest depth (Zhypo ≥ 100 km), these events play a crucial role in the seismic hazard assessment for several cities in northern South America (NoSAm), particularly those situated close to their epicenters. Moreover, these earthquakes are widely felt across the country, even hundreds of kilometers away from the nest. This paper assesses the performance of commonly used subduction intraslab Ground Motion Models (GMMs) in predicting the updated nest database gathered and processed by the Colombian Geological Survey. This analysis confirms the necessity of a partially non-ergodic ground motion model for estimating the spectral accelerations produced by Bucaramanga nest earthquakes in northern South America. The GMM presented herein is the first model developed specifically for seismic nests in the region and is conceived as a regionalization of the recent NGA-Sub Abrahamson and Gülerce (Earthq Spectra 38(4):2638−2681, 2022) GMM. Additionally, we provide estimates of the variance components at both the earthquake and station levels and compare the developed GMM with the spectral accelerations recorded for typical earthquakes, for the highest magnitude earthquake in the database, and for a recent Bucaramanga nest earthquake having a moment magnitude over 5. [ABSTRACT FROM AUTHOR]

Considering energy harvesting, the ergodic data rates for both in band full-duplex (FD) and half-duplex (HD) wireless communications were studied. The analytic expressions of downlink and uplink ergodic rates for the proposed system were first derived with independent and identically distributed (i.i.d.) Rayleigh fading link. It was revealed that the uplink data rate can be improved by decreasing the downlink data rate. Furthermore, the uplink/downlink data rates are also shown to be influenced by some significance parameters, for example, the power split parameter and signal-to-noise ratio (SNR) (i.e., PS/σ2) of each link. Additionally, unlike the HD, the proposed FD node is capable of harvesting energy during the communication process; however, this is at the cost of performance loss induced by the residual self-interference (RSI), which is caused by the essence of simultaneous uplink and downlink transmissions in a single frequency band. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case. [ABSTRACT FROM AUTHOR]

We prove that any strongly mixing action of a countable abelian group on a probability space has higher-order mixing properties. This is achieved via the utilization of $\mathcal R$ -limits, a notion of convergence which is based on the classical Ramsey theorem. $\mathcal R$ -limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP $^*$. While the main goal of this paper is to establish a universal property of strongly mixing actions of countable abelian groups, our results, when applied to ${\mathbb {Z}}$ -actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for ${\mathbb {Z}}$ -actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemerédi's theorem. We also demonstrate the versatility of $\mathcal R$ -limits by obtaining new characterizations of higher-order weak and mild mixing for actions of countable abelian groups. [ABSTRACT FROM AUTHOR]

A recent author's fractal fluid-dynamic dispersion theory in porous media has focused on the derivation of the associated nonergodic (or effective) macrodispersion coefficients by a 3-D stochastic Lagrangian approach. As shown by the present study, the Fickian (i.e., the asymptotic constant) component of a properly normalized version of these coefficients exhibits a clearly detectable minimum in correspondence with the same fractal dimension (d ≅ 1.7) that seems to characterize the diffusion-limited aggregation state of cells in advanced stages of cancerous lesion progression. That circumstance suggests that such a critical fractal dimension, which is also reminiscent of the colloidal state of solutions (and may therefore identify the microscale architecture of both living and non-living two-phase systems in state transition conditions) may actually represent a sort of universal nature imprint. Additionally, it suggests that the closed-form analytical solution that was provided for the effective macrodispersion coefficients in fractal porous media may be a reliable candidate as a physically-based descriptor of blood perfusion dynamics in healthy as well as cancerous tissues. In order to evaluate the biological meaningfulness of this specific fluid-dynamic parameter, a preliminary validation is performed by comparison with the results of imaging-based clinical surveys. Moreover, a multifractal extension of the theory is proposed and discussed in view of a perspective interpretative diagnostic utilization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

By 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 ×r$\times r$‐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 ×r$\times r$‐ and ×s$\times s$‐invariant;integer analogs of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection A∩B$A\cap B$ and the sumset A+B$A+B$ of ×r$\times r$‐ and ×s$\times s$‐invariant sets A$A$ and B$B$ when r$r$ and s$s$ are multiplicatively independent; anda description of the dimension of iterated sumsets A+A+⋯+A$A+A+\cdots +A$ for any ×r$\times r$‐invariant set A$A$.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 Lq$L^q$‐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. [ABSTRACT FROM AUTHOR]

In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving systems acted by Z d to the case of measure-preserving systems acted by Z d with zero entropy. As an application, we can build pointwise convergence of polynomial ergodic averages for K -systems acted by Z d. [ABSTRACT FROM AUTHOR]

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification. [ABSTRACT FROM AUTHOR]

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upper Banach density $d^*(E) = \delta > 0$ , we show, inter alia : (1) if $p(x) \in K[x]$ is an intersective polynomial (i.e., p has a root modulo m for every $m \in \mathcal{O}_K$) with $p(\mathcal{O}_K) \subseteq \mathcal{O}_K$ and $r, s \in \mathcal{O}_K$ are distinct and nonzero, then for any $\varepsilon > 0$ , there is a syndetic set $S \subseteq \mathcal{O}_K$ such that for any $n \in S$ , \begin{align*} d^* \left(\left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n)\} \subseteq E \right\} \right) > \delta^3 - \varepsilon. \end{align*} Moreover, if ${s}/{r} \in \mathbb{Q}$ , then there are syndetically many $n \in \mathcal{O}_K$ such that \begin{align*} d^* \left(\left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \right\} \right) > \delta^4 - \varepsilon; \end{align*} (2) if $\{p_1, \dots, p_k\} \subseteq K[x]$ is a jointly intersective family (i.e., $p_1, \dots, p_k$ have a common root modulo m for every $m \in \mathcal{O}_K$) of linearly independent polynomials with $p_i(\mathcal{O}_K) \subseteq \mathcal{O}_K$ , then there are syndetically many $n \in \mathcal{O}_K$ such that \begin{align*} d^* \left(\left\{ x \in \mathcal{O}_K \;:\; \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \right\} \right) > \delta^{k+1} - \varepsilon. \end{align*} These two results generalise and extend previous work of Frantzikinakis and Kra [ 21 ] and Franztikinakis [ 19 ] on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best [ 2 ] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg's correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl's equidistribution theorem for polynomials of several variables: (3) let $d, k, l \in \mathbb{N}$. Let $(X, \mathcal{B}, \mu, T_1, \dots, T_l)$ be an ergodic, connected $\mathbb{Z}^l$ -nilsystem. Let $\{p_{i,j} \;:\; 1 \le i \le k, 1 \le j \le l\} \subseteq \mathbb{Q}[x_1, \dots, x_d]$ be a family of polynomials such that $p_{i,j}\left(\mathbb{Z}^d \right) \subseteq \mathbb{Z}$ and $\{1\} \cup \{p_{i,j}\}$ is linearly independent over $\mathbb{Q}$. Then the $\mathbb{Z}^d$ -sequence $\left(\prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, \dots, \prod_{j=1}^l{T_j^{p_{k,j}(n)}}x \right)_{n \in \mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure. [ABSTRACT FROM AUTHOR]

Recently, the growth of the products of consecutive partial quotients a i (x) in the continued fraction expansion of a real number x was studied in connections with improvements to Dirichlet's theorem. In this paper, for a non-decreasing positive measurable function F (x 1 , ... , x m) and a function ϕ : ℕ → ℝ > 0 , we consider the set ℰ F (ϕ) = { x ∈ [ 0 , 1 ] : F (a n (x) , ... , a n + m − 1 (x)) ≥ ϕ (n)  for infinitely many  n ∈ ℕ } , and obtain its Lebesgue measure ℒ (ℰ F (ϕ)). As an application of our result, we reprove a theorem of Bakhtawar–Hussain–Kleinbock–Wang. We also consider the case when F (x 1 , ... , x m) = x 1 + ⋯ + x m . [ABSTRACT FROM AUTHOR]

This paper presents some numerical experiments in relation with the theoretical study of the ergodic short-term behaviour of discretizations of expanding maps done in P.-A. Guihéneuf and M. Monge, [Cramér distance and discretizations of circle expanding maps I: theory, (2022). arXiv 2206.07991]. Our aim is to identify the phenomena driving the evolution of the distance between the tth iterate of Lebesgue measure by the dynamics f and the tth iterate of the uniform measure on the grid of order N by the discretization on this grid. Based on numerical simulations we propose some conjectures on the effects of numerical truncation from the ergodic viewpoint. [ABSTRACT FROM AUTHOR]

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure. [ABSTRACT FROM AUTHOR]

We answer a question posed by Vershik regarding connections between quasi-similarity of dynamical systems and Kolmogorov entropy. We prove that all Bernoulli actions of a given countably infinite group are quasi-similar to each other. The existence of non-Bernoulli actions in the same quasi-similarity class is an open problem. A notion opposite to quasi-similarity is that of disjointness (or independence) of actions. Pinsker proved that a deterministic action is independent from an action with completely positive entropy. Using joinings, we obtain the following generalization of Pinsker's theorem: an action with zero -entropy (an invariant defined by Kirillov and Kushnirenko) and an action with completely positive -entropy are disjoint. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The velocity distribution of a classical gas of atoms in thermal equilibrium is the normal Maxwell distribution. It is well known that for sub-recoiled laser cooled atoms, Lévy statistics and deviations from usual ergodic behavior come into play. In a recent letter, we showed how tools from infinite ergodic theory describe the cool gas. Here, using the master equation, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of laser cooled atoms, recapitulating results obtained by Bertin and Bardou [Am. J. Phys. 76, 630 (2008)] using life-time statistics. We focus on the case where the laser trapping is strong, namely, the rate of escape from the velocity trap is R(v) ∝ |v|α for v → 0 and α > 1. We construct a machinery to investigate time averages of physical observables and their relation to ensemble averages. The time averages are given in terms of functionals of the individual stochastic paths, and here we use a generalization of Lévy walks to investigate the ergodic properties of the system. Exploring the energy of the system, we show that when α = 3, it exhibits a transition between phases where it is either an integrable or a non-integrable observable with respect to the infinite invariant measure. This transition corresponds to very different properties of the mean energy and to a discontinuous behavior of fluctuations. While the integrable phase is described by universal statistics and the Darling–Kac law, the more challenging case is the exploration of statistical properties of non-integrable observables. Since previous experimental work showed that both α = 2 and α = 4 are attainable, we believe that both phases could also be explored experimentally. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The semi-invertible version of Oseledets' multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman's subadditive ergodic theorem. [ABSTRACT FROM AUTHOR]

We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7, No. 1, 49–72 (1987)]. Namely, we show that if two Anosov flow of this kind are C0 conjugate, then they are Cr conjugate for some r ∈ [1, 2) or even C∞ conjugate under certain additional assumptions. This, e.g., applies to geodesic flows on compact Riemannian manifolds of 1/4-pinched negative sectional curvature. We can also use our result to recover Hamendstädt's marked length spectrum rigidity result for real hyperbolic manifolds. [ABSTRACT FROM AUTHOR]

This paper proposes a stochastic HIV/HTLV-I co-infection model incorporating the AIDS-related cancer cells and investigates its dynamical behaviours. The main methods are stochastic Lyapunov analysis and the ergodic theory. The existence and uniqueness of the global positive solution is proved as well as the stochastic ultimate boundedness. A unique stationary distribution is yielded. Furthermore, the sufficient conditions for the extinction of diseases are given. Finally, some numerical simulations are carried out to support the theoretical results. [ABSTRACT FROM AUTHOR]

Let (Ω, μ) be a measurable space with a σ-finite continuous measure, μ(Ω) = ∞. A linear operator T : L1(Ω) + L∞(Ω) → L1(Ω) + L∞(Ω) is called a Dunford–Schwartz operator if ∥T(f)∥1 ≤ ∥f∥1 (respectively, ∥T(f)∥∞ ≤ ∥f∥∞) for all f ∈ L1(Ω) (respectively, f ∈ L∞(Ω)). If {Tt}t≥0 is a strongly continuous in L1(Ω) semigroup of Dunford–Schwartz operators, then each operator At(f) = 1 t ∫ t 0 T s f d s ∈ L 1 Ω , f ∈ L1(Ω) has a unique extension to a Dunford–Schwartz operator, which is also denoted by At, t > 0. It is proved that in a completely symmetric space E(Ω) ⊈ L1 of measurable functions on (Ω, μ) the means At converge strongly as t → +∞ for each strongly continuous in L1(Ω) semigroup {Tt}t≥0 of Dunford–Schwartz operators if and only if the norm ∥·∥E(Ω) is order continuous. [ABSTRACT FROM AUTHOR]

We fully study the oriented rotatability exponents of solutions to homogeneous autonomous linear differential systems and establish that the strong and weak oriented rotatability exponents coincide for each solution to an autonomous system of differential equations. We also show that the spectrum of this exponent (i.e., the set of values of nonzero solutions) is naturally determined by the number-theoretic properties of the set of imaginary parts of the eigenvalues of the matrix of a system. This set (in contrast to the oscillation and wandering exponents) can contain other than zero values and the imaginary parts of the eigenvalues of the system matrix; moreover, the power of this spectrum can be exponentially large in comparison with the dimension of the space. In demonstration we use the basics of ergodic theory, in particular, Weyl's Theorem. We prove that the spectra of all oriented rotatability exponents of autonomous systems with a symmetrical matrix consist of a single zero value. We also establish relationships between the main values of the exponents on a set of autonomous systems. The obtained results allow us to conclude that the exponents of oriented rotatability, despite their simple and natural definitions, are not analogs of the Lyapunov exponent in oscillation theory. [ABSTRACT FROM AUTHOR]

We prove the equivalence of the power-law convergence rate in the -norm of ergodic averages for and actions and the same power-law estimate for the spectral measure of symmetric -dimensional parallelepipeds: for the degrees that are roots of some special symmetric polynomial in variables. Particularly, all possible range of power-law rates is covered for . [ABSTRACT FROM AUTHOR]

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang ['Hausdorff theory of dual approximation on planar curves', J. reine angew. Math. 740 (2018), 63–76] and Beresnevich and Velani ['A note on three problems in metric Diophantine approximation', in: Recent Trends in Ergodic Theory and Dynamical Systems , Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s -Hausdorff measure, where $s\in \mathbb R$ , to the more general g -Hausdorff measure, where g is a suitable class of dimension functions. [ABSTRACT FROM AUTHOR]

This article, titled "Historic Behavior in Nonautonomous Rock-Paper-Scissors Dynamics," explores the concept of historic behavior in dynamical systems. The author challenges the widely held belief that replicator equations in evolutionary game theory satisfy the "Folk Theorem of Evolutionary Game Theory." The paper proposes a class of replicator equations that exhibit historic behavior, where the time averages of the orbit do not converge. The article also presents some auxiliary results and provides a proof for the main theorem. The research was supported by ongoing institutional funding, and the author declares no conflicts of interest. [Extracted from the article]