Mean Ergodicity Research Articles

Ergodic Properties of Composition Semigroups on the Disc Algebra

Every semigroup $$\{\varphi _t\}_{t\ge 0}$$ of self-maps of the disc defines a semigroup $$\{C_{\varphi _t}\}_{t\ge 0}$$ of compositions operators on the space of holomorphic functions on the disc. We characterize the (uniform) mean ergodicity (in the sense of continuous means) and the asymptotic behaviour of these operators when they define a $$C_0$$ -semigroup on the disc algebra, in terms of the Denjoy–Wolff point and the associated planar domain in the sense of Berkson and Porta. Finally we deal with the case of Hardy and Bergman spaces.

On typical encodings of multivariate ergodic sources

We show that the typical coordinate-wise encoding of multivariate ergodic source into prescribed alphabets has the entropy profile close to the convolution of the entropy profile of the source and the modular polymatroid that is determined by the cardinalities of the output alphabets. We show that the proportion of the exceptional encodings that are not close to the convolution goes to zero doubly exponentially. The result holds for a class of multivariate sources that satisfy asymptotic equipartition property described via the mean fluctuation of the information functions. This class covers asymptotically mean stationary processes with ergodic mean, ergodic processes, irreducible Markov chains with an arbitrary initial distribution. We also proved that typical encodings yield the asymptotic equipartition property for the output variables. These asymptotic results are based on an explicit lower bound of the proportion of encodings that transform a multivariate random variable into a variable with the entropy profile close to the suitable convolution.

Explicit solution of Rice/Lalanne peak probability distribution for statistical fatigue assessment in the frequency domain

A closed-form explicit solution for statistical fatigue life assessment under broad-band stochastic loading is proposed herein. Obtained solution is valid for stationary, random and ergodic zero mean Gaussian stress process. It is based on Rice/Lalanne peak stress probability distribution function. The solution is given in terms of special Gaussian HyperGeometric function which accurately describes the transcendent error weighting function between the Rayleigh (i.e. narrow-band) and Gaussian (i.e. broad-band) contribution of the random vibration fatigue model. HyperGeometric-based solution is more accurate compared to other approximate methods based on the peak distribution from literature. Fatigue benchmark test comparison is performed in time and frequency domains with very good match-up reported.

Normalized concentrating solutions to nonlinear elliptic problems

We prove the existence of solutions (λ,v)∈R×H1(Ω) of the elliptic problem{−Δv+(V(x)+λ)v=vp in Ω,v>0,∫Ωv2dx=ρ. Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L2(Ω). Here Ω is either the whole space RN or a bounded smooth domain of RN, in which case we assume V≡0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, 1<p<N+2N−2 if N≥3 and p>1 if N=1,2. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of Ω as the prescribed mass ρ is either small (when p<1+4N) or large (when p>1+4N) or it approaches some critical threshold (when p=1+4N).

Operators acting in sequence spaces generated by Dual Banach spaces of discrete Cesàro spaces

The dual spaces $d(p), 1 \lt p \lt \infty ,$ of the discrete Cesaro (Banach) spaces $\mathrm{ces} (q)$, $1 \lt q \lt \infty ,$ were studied by G. Bennett, A. Jagers and others. These (reflexive) dual Banach spaces induce the non-normable Frechet spaces $d (p+) := \bigcap_{r \gt p} d (r), $ for $1 \leq p \lt \infty ,$ and the (LB)-spaces $d (p-) := \bigcup_{ 1 \lt r \lt p } d (r), $ for $ 1 \lt p \leq \infty ,$ recently introduced and investigated in [11]. Here a detailed study is made of various aspects, such as the spectrum, continuity, compactness, mean ergodicity and supercyclicity of the Cesaro operator, multiplication operators and inclusion operators when they act on (and between) such spaces.

Dynamics of weighted composition operators on weighted Banach spaces of entire functions

We study the dynamics of the weighted composition operator Cw,φ on weighted Banach spaces of entire functions Hv(C) and Hv0(C). We characterize the continuity and compactness of the operator and, in the case of affine symbols φ(z)=az+b,a,b∈C, and exponential weights, we analyze when the operator is power bounded, (uniformly) mean ergodic and hypercyclic. Continuous weighted composition operators when |a|=1 are just multiples of composition operators λCφλ∈C. When |a|<1, we consider as a multiplier w the product of a polynomial by an exponential function. For multiples of composition operators, we get a complete characterization of power boundedness and mean ergodicity and we study the hypercyclicity in terms of λ. An example of a power bounded but not mean ergodic operator on Hv0(C) is provided. For the case of composition operators, we obtain the spectrum and a complete characterization of the dynamics.

A partially inexact ADMM with o(1/n) asymptotic convergence rate, 𝒪(1/n) complexity, and immediate relative error tolerance

In this work, we propose a new partially inexact Alternating Direction Method of Multipliers (ADMM) with relative error tolerance. This method departs from previous semi-inexact variants of the ADMM by allowing the second subproblem to be solved inexactly, instead of the first one. Relative error tolerance criteria for inexact ADMM methods require, to be verified, the solution of the two proximal subproblems in each iteration; therefore, by allowing for inexactness in the approximate solutions of the second subproblem, the error criterion is immediately testable during the computation of that approximate solution and no backtracking is required. In this sense, the proposed method uses an immediate error criterion. We also establish iteration complexity and asymptotic convergence rate for a modified ergodic mean, with respect to the residuals in Karush–Kuhn–Tucker conditions. As far as we know, this is the first asymptotic convergence rate, with respect to this error measure, for ADMM applied to a general problem.

Ganikhodjaev’s Conjecture on Mean Ergodicity of Quadratic Stochastic Operators

A linear stochastic (Markov) operator is a positive linear contraction which preserves the simplex. A quadratic stochastic (nonlinear Markov) operator is a positive symmetric bilinear operator that preserves the simplex. The ergodic theory studies the long term average behavior of systems evolving in time. The classical mean ergodic theorem asserts that the arithmetic average of the linear stochastic operator always converges to some linear stochastic operator. While studying the evolution of the population system, S. Ulam conjectured the mean ergodicity of quadratic stochastic operators. However, M. Zakharevich showed that Ulam’s conjecture is false in general. Later, N. Ganikhodjaev and D. Zanin have generalized Zakharevich’s example in the class of quadratic stochastic Volterra operators. Afterward, N. Ganikhodjaev made a conjecture that Ulam’s conjecture is true for properly quadratic stochastic non-Volterra operators. In this paper, we provide counterexamples to Ganikhodjaev’s conjecture on mean ergodicity of quadratic stochastic operators acting on the higher dimensional simplex.

Operators acting in the dual spaces of discrete Cesàro spaces

The discrete Cesaro (Banach) sequence spaces $$ {{\text {ces}}}(r), 1< r < \infty ,$$ have been thoroughly investigated for over 45 years. Not so for their dual spaces $$ d (s) \cong ( {{\text {ces}}}(r))', $$ with $$ \frac{1}{s} + \frac{1}{r} = 1 ,$$ which are somewhat unwieldy. Our aim is to undertake a further study of the spaces d(s) and of various operators acting between these spaces. It is shown that $$ d (s) \subseteq d (t)$$ whenever $$ s \le t ,$$ with the inclusion being compact if $$ s< t .$$ The classical Cesaro operator C is continuous from d(s) into d(t) precisely when $$ s \le t $$ and compact precisely when $$ s < t .$$ Moreover, C even maps the larger space $$ {{\text {ces}}}(s)$$ continuously into d(s). This is a consequence of the Hardy–Littlewood maximal theorem and the remarkable property, for each $$ 1< s < \infty ,$$ that $$ x \in \mathbb {C}^{\mathbb {N}} $$ satisfies $$ C (C (| x| )) \in d (s)$$ if and only if $$ C (| x | ) \in d (s).$$ These results are used to analyze the spectrum and to determine the norm and the mean ergodicity of C acting in d(s). Similar properties for multiplier operators are also treated.

Iterated Means Dichotomy for Discrete Dynamical Systems

In this paper, we discuss a dichotomy of iterated means for a compact discrete dynamical system acting on a finite dimensional space. As an application, we also study the mean ergodicity of non-homogeneous Markov chains.

Ergodic properties of composition operators on Banach spaces of analytic functions

The composition operators defined on little Bloch spaces, Bergman spaces, Hardy spaces or little weighted Bergman spaces of infinite type, when well defined, are shown to be mean ergodic if and only if they are power bounded if and only if the symbol has an interior fixed point. For these operators uniform mean ergodicity is equivalent to quasicompactness in the sense of Yosida and Kakutani.

Ergodic properties of Markov semigroups in von Neumann algebras

We investigate ergodic properties of Markov semigroups in von Neumann algebras with the help of the notion of constrictor, which expresses the idea of closeness of the orbits of the semigroup to some set, as well as the notion of generalised averages, which generalises to arbitrary abelian semigroups the classical notions of Ces`aro, Borel, or Abel means. In particular, mean ergodicity, asymptotic stability, and structure properties of the fixed-point space are analysed in some detail.

Noncommutative strong maximals and almost uniform convergence in several directions

AbstractOur first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the$L_p$-norm of the$\limsup $of a sequence of operators as a localized version of a$\ell _\infty /c_0$-valued$L_p$-space. In particular, our main result gives a strong$L_1$-estimate for the$\limsup $—as opposed to the usual weak$L_{1,\infty }$-estimate for the$\mathop {\mathrm {sup}}\limits $—with interesting consequences for the free group algebra.Let$\mathcal{L} \mathbf{F} _2$denote the free group algebra with$2$generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside$L_1(\mathcal{L} \mathbf{F} _2)$for which the free Poisson semigroup converges to the initial data. Currently, the best known result is$L \log ^2 L(\mathcal{L} \mathbf{F} _2)$. We improve this result by adding to it the operators in$L_1(\mathcal{L} \mathbf{F} _2)$spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative$\limsup $together with new transference techniques.We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak$(\Phi ,\Phi )$inequality—as opposed to weak$(\Phi ,1)$—for noncommutative multiparametric martingales and$\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$. This logarithmic power is an$\varepsilon $-perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

Generalized Dobrushin ergodicity coefficient and uniform ergodicities of Markov operators

In this paper the stability and the perturbation bounds of Markov operators acting on abstract state spaces are investigated. Here, an abstract state space is an ordered Banach space where the norm has an additivity property on the cone of positive elements. We basically study uniform ergodic properties of Markov operators by means of so-called a generalized Dobrushin's ergodicity coefficient. This allows us to get several convergence results with rates. Some results on quasi-compactness of Markov operators are proved in terms of the ergodicity coefficient. Furthermore, a characterization of uniformly $P$-ergodic Markov operators is given which enable us to construct plenty examples of such types of operators. The uniform mean ergodicity of Markov operators is established in terms of the Dobrushin ergodicity coefficient. The obtained results are even new in the classical and quantum settings

Ganikhodjaev's conjecture on mean ergodicity of quadratic stochastic operators

Ganikhodjaev's conjecture on mean ergodicity of quadratic stochastic operators

Mean ergodicity of composition operators on Hardy space

In this article, we investigate the mean ergodicity of composition operators acting on the Hardy space $$H^p({\mathbb {D}})$$ , $$1\le p<\infty $$ . Specifically, a composition operator $$C_\varphi $$ acting on $$H^p({\mathbb {D}})$$ is mean ergodic if and only if $$\varphi $$ has an interior fixed point, in which case $$C_\varphi $$ is uniformly mean ergodic if and only if $$\varphi $$ is an elliptic automorphism of finite order or a non-automorphism that is not inner.

Toward practical approaches for ergodicity analysis

It is of importance to perform hydrological forecast using a finite hydrological time series. Most time series analysis approaches presume a data series to be ergodic without justifying this assumption. To our knowledge, there are no methods available for test of ergodicity to date. This paper presents a practical approach to analyze the mean ergodic property of hydrological processes by means of augmented Dickey Fuller test, Mann-Kendall trend test, a radial basis function neural network, and the assessment methods derived from the definition of ergodicity. The mean ergodicity of precipitation processes at Newberry, MI, USA, is analyzed using the proposed approach. The results indicate that the precipitations of January, May, and July in Newberry are highly likely to have ergodic property, the precipitations of February, and October through December have tendency toward mean ergodicity, and the precipitations of all the other months are non-ergodic.

The primitive spectrum of a semigroup of Markov operators

For a semigroup $$\mathcal {S}$$ of Markov operators on a space of continuous functions, we use $$\mathcal {S}$$-invariant ideals to describe qualitative properties of $$\mathcal {S}$$ such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive$$\mathcal {S}$$-ideals and endow the space of those ideals with an appropriate topology. This approach is inspired by the representation theory of C*-algebras and can be adapted to our dynamical setting. In the particularly important case of Koopman semigroups, we characterize the centers of attraction of the underlying dynamical system in terms of the invariant ideal structure of $$\mathcal {S}$$.

A Benchmarked Evaluation of a Selected CapitalCube Interval-Scaled Market Performance Variable

Context In this fifth analysis of the CapitalCube™ Market Navigation Platform[CCMNP], the focus is on the CaptialCube Closing Price Latest [CCPL] which, is an Interval Scaled Market Performance [ISMP] variable that seems, a priori, the key CCMNP information for tracking the price of stocks traded on the S&amp;P500. This study follows on the analysis of the CCMNP’s Linguistic Category MPVs [LCMPV] where it was reported that the LCMPV were not effective in signaling impending Turning Points [TP] in stock prices. Study Focus As the TP of an individual stock is the critical point in the Panel and was used previously in the evaluation of the CCMNP, this study adopts the TP as the focal point in the evaluation montage used to determine the market navigation utility of the CCPL. This study will use the S&amp;P500 Panel in an OLS Time Series [TS] two-parameter linear regression context: Y[S&amp;P500] = X[TimeIndex] as the Benchmark for the performance evaluation of the CCPL in the comparable OLS Regression: Y[S&amp;P500] = X[CCPL]. In this regard, the inferential context for this comparison will be the Relative Absolute Error [RAE] using the Ergodic Mean Projection [termed the Random Walk[RW]] of the matched-stock price forecasts three periods after the TP. Results Using the difference in the central tendency of the RAEs as the effect-measure, the TS: S&amp;P Panel did not test to be different from the CCPL-arm of the study; further neither outperformed the RW; all three had Mean and Median RAEs that were greater than 1.0—the standard cut-point for rationalizing the use of a particular forecasting model. Additionally, an exploratory analysis used these REA datasets blocked on: (i) horizons and (ii) TPs of DownTurns &amp; UpTurns; this analysis identified interesting possibilities for further analyses.

Long-term analysis of positive operator semigroups via asymptotic domination

We consider positive operator semigroups on ordered Banach spaces and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $$\mathcal {T}$$ are inherited by other semigroups which are asymptotically dominated by $$\mathcal {T}$$ . Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces.