Fully oscillating sequences and weighted multiple ergodic limit

We first develop a theory of conditional expectations for random variables with values in a complete metric space [Formula: see text] equipped with a contractive barycentric map [Formula: see text], and then give convergence theorems for martingales of [Formula: see text]-conditional expectations. We give the Birkhoff ergodic theorem for [Formula: see text]-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the [Formula: see text]-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on [Formula: see text]. Finally, the large deviation property of [Formula: see text]-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

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Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics

Journal of Nonlinear Science ◽

10.1007/s00332-020-09652-7 ◽

2020 ◽

Vol 30 (6) ◽

pp. 3321-3366

Author(s):

Luzie Helfmann ◽

Enric Ribera Borrell ◽

Christof Schütte ◽

Péter Koltai

Keyword(s):

Finite Time ◽

Social Dynamics ◽

Transition Path ◽

Time Dynamics ◽

Transition Path Theory ◽

Periodically Driven ◽

Statistical Behavior ◽

Ergodic Limit ◽

Dynamical Regimes ◽

Time Systems

Abstract Given two distinct subsets A, B in the state space of some dynamical system, transition path theory (TPT) was successfully used to describe the statistical behavior of transitions from A to B in the ergodic limit of the stationary system. We derive generalizations of TPT that remove the requirements of stationarity and of the ergodic limit and provide this powerful tool for the analysis of other dynamical scenarios: periodically forced dynamics and time-dependent finite-time systems. This is partially motivated by studying applications such as climate, ocean, and social dynamics. On simple model examples, we show how the new tools are able to deliver quantitative understanding about the statistical behavior of such systems. We also point out explicit cases where the more general dynamical regimes show different behaviors to their stationary counterparts, linking these tools directly to bifurcations in non-deterministic systems.

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On instability of the ergodic limit theorems with respect to small violations of algorithmic randomness

2011 IEEE International Symposium on Information Theory Proceedings ◽

10.1109/isit.2011.6033795 ◽

2011 ◽

Cited By ~ 1

Author(s):

Vladimir V'yugin

Keyword(s):

Limit Theorems ◽

Algorithmic Randomness ◽

Ergodic Limit

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Optimization of Generalized Nanostructures for Light Trapping Beyond the Ergodic Limit

Renewable Energy and the Environment ◽

10.1364/pv.2013.pt2c.4 ◽

2013 ◽

Author(s):

Peng Wang ◽

Bing Shen ◽

Rajesh Menon

Keyword(s):

Light Trapping ◽

Ergodic Limit

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Exact regularity and the cohomology of tiling spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000611 ◽

2011 ◽

Vol 31 (6) ◽

pp. 1819-1834 ◽

Cited By ~ 8

Author(s):

LORENZO SADUN

Keyword(s):

Convergence Rate ◽

One Dimension ◽

One Dimensional ◽

Uniform Estimates ◽

Tiling Spaces ◽

Ergodic Limit

AbstractExact regularity was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider the analog of exact regularity for arbitrary tiling spaces. Let T be a d-dimensional repetitive tiling, and let Ω be its hull. If Ȟd(Ω,ℚ)=ℚk, then there exist k patches each of whose appearances governs the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling T comes from a substitution, then we can quantify that convergence rate. If T is also one dimensional, we put constraints on the measure of any cylinder set in Ω.

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Asymptotic behavior of a Feller evolution family involved in the Fisher-Wright model

Advances in Applied Probability ◽

10.1239/aap/1222868184 ◽

2008 ◽

Vol 40 (3) ◽

pp. 734-758 ◽

Cited By ~ 1

Author(s):

Adam Bobrowski

Keyword(s):

Asymptotic Behavior ◽

Joint Distribution ◽

Transition Probabilities ◽

Time Varying ◽

Asymptotically Stable ◽

Feller Process ◽

The Past ◽

Ergodic Limit ◽

Relative Differences ◽

Limit Form

We study the evolution in time of the joint distribution of a pair of Feller processes, related by the fact that some random time ago they were identical, evolving as a single Feller process; from that time on, they began to evolve independently, conditional on a state at the time of split, according to the same Feller transition probabilities. Such processes are involved in the Fisher-Wright model: the distribution of the time counted backwards from the present to the time of split in the past is a function of deterministic but time-varying effective size 2N of the population from which the two processes are sampled. In terms of a corresponding family of Feller operators, assuming asymptotic stability or ergodicity of the process of mutation, we find the limit form of the distribution of such pairs of processes sampled from decaying, asymptotically constant, and growing populations. In the case where mutation is not asymptotically stable or ergodic, limit distributions are found for the distribution of relative differences.

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Delone dynamical systems and spectral convergence

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.116 ◽

2018 ◽

Vol 40 (6) ◽

pp. 1510-1544 ◽

Cited By ~ 3

Author(s):

SIEGFRIED BECKUS ◽

FELIX POGORZELSKI

Keyword(s):

Dynamical Systems ◽

Systems Approach ◽

Special Focus ◽

Bounded Operators ◽

Special Situation ◽

Ergodic Limit ◽

Spectral Convergence ◽

Recent Developments ◽

Delone Sets ◽

Uniquely Ergodic

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

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