scholarly journals ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS

2009 ◽  
Vol 09 (04) ◽  
pp. 635-655 ◽  
Author(s):  
H. BRUIN ◽  
M. NICOL ◽  
D. TERHESIU
Keyword(s):  
Dynamical System ◽  
Common Factor ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Original System ◽  
Return Time ◽  
Tail Behavior ◽  
Necessary And Sufficient ◽  
Young Towers ◽  
Measure Preserving Transformations

For a σ-finite measure preserving dynamical system (X, μ, T), we formulate necessary and sufficient conditions for a Young tower (Δ, ν, F) to be a (measure theoretic) extension of the original system. Because F is pointwise dual ergodic by construction, one immediate consequence of these conditions is that the Darling–Kac theorem carries over from F to T. One advantage of the Darling–Kac theorem in terms of Young towers is that sufficient conditions can be read off from the tail behavior and we illustrate this with relevant examples. Furthermore, any two Young towers with a common factor T, have return time distributions with tails of the same order.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

The Pointwise Ergodic Theorem for Transformations whose Orbits contain or are contained in the Orbits of a Measure-Preserving Transformation

1982 ◽  
Vol 34 (6) ◽  
pp. 1303-1318 ◽  
Author(s):  
John C. Kieffer ◽  
Maurice Rahe
Keyword(s):  
Information Theory ◽  
Ergodic Theory ◽  
Probability Space ◽  
Sufficient Conditions ◽  
Measurable Transformation ◽  
Input And Output ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Measure Preserving Transformations

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

Unique Extension and Product Measures

1967 ◽  
Vol 19 ◽  
pp. 757-763 ◽  
Author(s):  
Norman Y. Luther
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Product Measure ◽  
Unique Extension ◽  
Product Measures ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Finite Measures

Following (2) we say that a measure μ on a ring is semifinite ifClearly every σ-finite measure is semifinite, but the converse fails.In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

scholarly journals Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

2011 ◽  
Vol 43 (3) ◽  
pp. 688-711 ◽  
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

Multi-Variable Control of an Integrated Propulsion and Airframe System

1995 ◽  
Vol 209 (3) ◽  
pp. 179-190
Author(s):  
R A Perez
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Closed Loop ◽  
Integrated Control ◽  
Sufficient Conditions ◽  
Turbofan Engine ◽  
Variable Control ◽  
Control Scheme ◽  
Necessary And Sufficient ◽  
Control Methodology

The development of an integrated control scheme to enhance the performance of a generic interconnected multi-variable dynamical system, consisting of a turbofan engine and an airframe, in the presence of predominantly destructive dynamical interactions over the flight envelope is considered in this paper. The control scheme consists of two components: a simple static forward loop or feedback loop precompensator to improve the interactions followed by a forward or feedback loop controller to improve the performance. The system must be tolerant to soft and hard output sensor failures by means of analytic redundancy only. A control methodology to satisfy the above specifications is presented here. Necessary and sufficient conditions are presented in order to achieve a stable closed-loop performance of the overall system by tuning every loop separately, that is decentralized stability.

scholarly journals A constructive view on ergodic theorems

2006 ◽  
Vol 71 (2) ◽  
pp. 611-623
Author(s):  
Bas Spitters
Keyword(s):  
Ergodic Theorem ◽  
Sufficient Conditions ◽  
Ergodic Measure ◽  
Constructive Mathematics ◽  
Ergodic Theorems ◽  
Invariant Functions ◽  
Almost Everywhere ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Measure Preserving Transformations

AbstractLet T be a positive L1-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics.(1) The projection in L2, on the space of invariant functions exists:(2) The sequence (Tn)n∈N Cesáro-converges in the L2 norm:(3) The sequence (Tn)n∈N Cesáro-converges almost everywhere.Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations.This answers a question posed by Bishop.

scholarly journals Joint ergodicity for group actions

1988 ◽  
Vol 8 (3) ◽  
pp. 351-364 ◽  
Author(s):  
Vitaly Bergelson ◽  
Joseph Rosenblatt
Keyword(s):  
Probability Space ◽  
Amenable Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compact Sets ◽  
Weakly Mixing ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Continuous Representations ◽  
Measure Preserving Transformations

AbstractLet T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) whereconverges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.

Generation of chaotic attractors without equilibria via piecewise linear systems

2017 ◽  
Vol 28 (01) ◽  
pp. 1750008 ◽  
Author(s):  
R. J. Escalante-González ◽  
E. Campos-Cantón
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic Attractor ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Chaotic Attractors ◽  
Necessary And Sufficient ◽  
Switched Dynamical System ◽  
Theoretical Results

In this paper, we present a mechanism of generation of a class of switched dynamical system without equilibrium points that generates a chaotic attractor. The switched dynamical systems are based on piecewise linear (PWL) systems. The theoretical results are formally given through a theorem and corollary which give necessary and sufficient conditions to guarantee that a linear affine dynamical system has no equilibria. Numerical results are in accordance with the theory.

scholarly journals Interchanging a limit and an integral: necessary and sufficient conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Takashi Kamihigashi
Keyword(s):  
Measure Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Integrable Functions ◽  
Finite Measure Space ◽  
Pointwise Limit ◽  
Necessary And Sufficient

AbstractLet $\{f_{n}\}_{n \in \mathbb {N}}$ { f n } n ∈ N be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ ( Ω , F , μ ) . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ lim n ↑ ∞ f n exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$ lim n ↑ ∞ ∫ f n d μ = ∫ lim n ↑ ∞ f n d μ .

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