scholarly journals WHAT IS...an Ergodic Transformation?

2016 ◽  
Vol 63 (01) ◽  
pp. 26-27
Author(s):  
Cesar E. Silva
Keyword(s):  
Ergodic Transformation

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

scholarly journals Geometry and dynamics of admissible metrics in measure spaces

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Anatoly Vershik ◽  
Pavel Zatitskiy ◽  
Fedor Petrov
Keyword(s):  
Invariant Measure ◽  
Wide Class ◽  
Lebesgue Space ◽  
Measure Space ◽  
Asymptotic Properties ◽  
Ergodic Transformation ◽  
Measure Spaces ◽  
Admissible Metric ◽  
Discreteness Criterion ◽  
Uniformly Bounded

AbstractWe study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

Sweeping Out Properties of Operator Sequences

1997 ◽  
Vol 49 (1) ◽  
pp. 3-23 ◽  
Author(s):  
Mustafa A. Akcoglu ◽  
Dzung M. Ha ◽  
Roger L. Jones
Keyword(s):  
Banach Spaces ◽  
Orthogonal Projection ◽  
Joint Distribution ◽  
Probability Space ◽  
Ergodic Transformation ◽  
Technical Result

AbstractLet Lp = Lp(X, μ), 1 ≤ p ≤ ∞, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let (T1, . . . ,TK) be L2-contractions. Let 0 < ε < δ ≤ 1. Call a function f a δ-spanning function if ‖f‖2 = 1 and if ‖Tkf - Qk-1Tkf‖2 ≥ δ for each k = 1, . . . ,K, where Q0 = 0 and Qk is the orthogonal projection on the subspace spanned by (T1f , . . . ,Tkf). Call a function h a (δ, ε) -sweeping function if ‖h‖∞ ≤ 1, ‖h‖1 < ε, and if max1≤k≤K|Tkh| > δ-ε on a set of measure greater than 1 - ε. The following is the main technical result, which is obtained by elementary estimates. There is an integer K = K(δ, ε) ≥ 1 such that if f is a δ-spanning function, and if the joint distribution of (f , T1f , . . . ,TKf) is normal, then h = ((fΛM)Ꮩ(-M)/M is a (δ, ε)-sweeping function, for some M > 0. Furthermore, if Tks are the averages of operators induced by the iterates of ameasure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence (Ti) of these averages.Assume that for each K ≤ 1 there is a subsequence (Ti1 , . . . ,Tik) of length K, and a δ-spanning function fK for this subsequence. Then for each ε > 0 there is a function h, 0 ≥ h ≥ 1, ‖h‖1 < ε, such that lim supi Tih ≤ δ a.e.. Another application of the main result gives a refinement of a part of Bourgain’s “Entropy Theorem”, resulting in a different, self contained proof of that theorem.

An ergodic transformation with trivial Kakutani centralizer

1992 ◽  
Vol 12 (3) ◽  
pp. 459-478 ◽  
Author(s):  
Adam Fieldsteel ◽  
Daniel J. Rudolph
Keyword(s):  
Dynamical System ◽  
Lebesgue Space ◽  
Ergodic Transformation ◽  
Orbit Equivalence ◽  
Factor Map ◽  
Image Position ◽  
Factor Maps

AbstractGeneralizing from the centralizer of a measure-preserving dynamical system (X, ℬ, μ,T), one defines the Kakutani centralizerKC(T) of all ‘even Kakutani factor maps’ ϕ fromTto itself. Such a ϕ is a composition φ2ϕ1of an even Kakutani orbit equivalence ϕ1and a factor map ϕ2. We construct here an ergodicTacting on a nonatomic Lebesgue space (X,ℬ,μ) with the property that any ϕ ∈ KC(T) is invertible and of the formAll invertible maps of this form are automatically in KC(T) and hence for thisTthe Kakutani centralizer is as small as possible.

scholarly journals Rock-Paper-Scissors Lattice Model

2020 ◽  
Vol 16 (4) ◽  
pp. 400-402
Author(s):  
Nasir Ganikhodjaev ◽  
Pah Chin Hee
Keyword(s):  
Lattice Model ◽  
Statistical Physics ◽  
Lattice Models ◽  
Evolutionary Games ◽  
Payoff Matrix ◽  
Ergodic Transformation ◽  
Translation Invariant ◽  
Periodic Gibbs Measure ◽  
Quadratic Stochastic Operators ◽  
Player Game

In this work, we introduce Rock-Paper-Scissors lattice model on Cayley tree of second order generated by Rock-Paper-Scissors game. In this strategic 2-player game, the rule is simple: rock beats scissors, scissors beat paper, and paper beats rock. A payoff matrix  of this game is a skew-symmetric. It is known that quadratic stochastic operator generated by this matrix is non-ergodic transformation. The Hamiltonian of Rock-Paper-Scissors Lattice Model is defined by this skew-symmetric payoff matrix . In this paper, we discuss a connection between three fields of research: evolutionary games, quadratic stochastic operators, and lattice models of statistical physics. We prove that a phase diagram of the Rock-Paper-Scissors model consists of translation-invariant and periodic Gibbs measure with period 3.

scholarly journals Coboundary equations of eventually expanding transformations

2003 ◽  
Vol 67 (1) ◽  
pp. 39-50
Author(s):  
Young-Ho Ahn
Keyword(s):  
Measurable Function ◽  
Finite Subgroup ◽  
Unit Interval ◽  
Continuous Measure ◽  
Ergodic Transformation ◽  
Skew Product ◽  
Markov Condition ◽  
Absolutely Continuous ◽  
Circle Group ◽  
Absolutely Continuous Measure

Let T be an eventually expansive transformation on the unit interval satisfying the Markov condition. The T is an ergodic transformation on (X, ß, μ) where X = [0, 1), ß is the Borel σ-algebra on the unit interval and μ is the T invariant absolutely continuous measure. Let G be a finite subgroup of the circle group or the whole circle group and φ: X → G be a measurable function with finite discontinuity points. We investigate ergodicity of skew product transformations Tφ on X × G by showing the solvability of the coboundary equation φ(x) g (Tx) = λg (x), |λ| = 1. Its relation with the uniform distribution mod M is also shown.

LIPSCHITZ–ERGODIC TRANSFORMATION

1988 ◽  
Vol 8 (1) ◽  
pp. 113-120
Author(s):  
Zou Ying
Keyword(s):  
Ergodic Transformation
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