Conjugates of Infinite Measure Preserving Transformations

1988 ◽  
Vol 40 (3) ◽  
pp. 742-749
Author(s):  
S. Alpern ◽  
J. R. Choksi ◽  
V. S. Prasad
Keyword(s):  
Conjugacy Class ◽  
Ergodic Theory ◽  
Lebesgue Space ◽  
Finite Measure ◽  
Infinite Measure ◽  
Ergodic Automorphism ◽  
Measurable Set ◽  
Image Position ◽  
Measure Preserving Transformations

In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we proveTHEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such thatThen there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani ([7], Theorem 6).Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.

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.

Coalescence of circle extensions of measure-preserving transformations

1992 ◽  
Vol 12 (4) ◽  
pp. 769-789 ◽  
Author(s):  
Mariusz Lemańczyk ◽  
Pierre Liardet ◽  
Jean-Paul Thouvenot
Keyword(s):  
Group Extension ◽  
Ergodic Transformations ◽  
Ergodic Automorphism ◽  
Special Cases ◽  
Image Position ◽  
Group Law ◽  
Measure Preserving Transformations

AbstractWe prove that for each ergodic automorphism T:(X, ℬ, μ)→(X, ℬ, μ) for which we can find an element S∈C(T) such that the corresponding Z2-action (S, T) on (X, ℬ, μ) is free, there exists a circle valued cocycle φ such that the group extension Tφ is ergodic but is not coalescent. In particular, the existence of such a cocycle is proved for all ergodic rigid automorphisms. As a corollary, in the class of ergodic transformations of [0,1) × [0,1) given byfor each irrational α we find φ such that Tφ is not coalescent. In some special cases the group law of the centralizer is given.

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.

On (L1)* for General Measure Spaces

1963 ◽  
Vol 6 (2) ◽  
pp. 211-229 ◽  
Author(s):  
H. W. Ellis ◽  
D. O. Snow
Keyword(s):  
Measurable Function ◽  
Measure Space ◽  
Finite Measure ◽  
Signed Measure ◽  
General Measure ◽  
Measure Spaces ◽  
Arbitrary Measure ◽  
Image Position ◽  
Nikodym Theorem

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

scholarly journals Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

2020 ◽  
Vol 150 (5) ◽  
pp. 2682-2718 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Antonio J. Fernández
Keyword(s):  
Bounded Domain ◽  
Lebesgue Space ◽  
Fractional Laplacian ◽  
Existence Result ◽  
Existence Results ◽  
Real Parameter ◽  
Nonlocal Diffusion ◽  
Smooth Bounded Domain ◽  
Image Position ◽  
Regularity Results

AbstractLet$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form $$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$ where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, $\mu \in L^{\infty}$ and $\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by $$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$ Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

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.

Piecewise isometries, uniform distribution and 3log 2−π2/8

2011 ◽  
Vol 32 (6) ◽  
pp. 1862-1888 ◽  
Author(s):  
YITWAH CHEUNG ◽  
AREK GOETZ ◽  
ANTHONY QUAS
Keyword(s):  
Phase Space ◽  
Uniform Distribution ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Full Measure ◽  
Infinite Measure ◽  
Important Family ◽  
Large Numbers ◽  
Piecewise Isometries ◽  
Measure Preserving Transformations

AbstractWe use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

scholarly journals Inequalities relating different definitions of discrepancy

1974 ◽  
Vol 17 (1) ◽  
pp. 81-87 ◽  
Author(s):  
C. J. Smyth
Keyword(s):  
Convex Sets ◽  
Measurable Set ◽  
Image Position

LetP1, P2…, pN be N points in the unit s-dimensional closed square Q = [0, 1]s. For any measurable set S ⊆ Q, we define δ(S), the discrepancy of S, by , where V(S) is the s-dimensional volume of S, and n(S), is the number of indices i for which pi∈S. Let , where the supermum is taken over all s-balls B ∈ Q, and , the supermum in this case being taken over all convex sets C ∈ Q. Clearly Dc ≧ Dk. In this paper we establish Theorem , where φ1is a constant depending only on s.

Bourgain Algebras of Spaces of n-Harmonic Functions in the Unit Polydisk

1996 ◽  
Vol 39 (3) ◽  
pp. 284-293
Author(s):  
Keiji Izuchi ◽  
Kazuhiro Kasuga ◽  
Yasuo Matsugu
Keyword(s):  
Characteristic Function ◽  
Lebesgue Space ◽  
Harmonic Functions ◽  
Unit Polydisk ◽  
Image Position

AbstractLet h∞(Dn) denote the space of all bounded n-harmonic functions on the unit polydisk Dn of Cn. In this paper we prove that the Bourgain algebra h∞(Dn)b and h∞(Dn)bb relative to the Lebesgue space L∞(Dn) are of the following forms:Here V(Dn) is the space of those functions such that , where denotes the characteristic function of a subset E of Dn.

Ratio and Stochastic Ergodic Theorems for Superadditive Processes

1979 ◽  
Vol 31 (2) ◽  
pp. 441-447 ◽  
Author(s):  
Humphrey Fong
Keyword(s):  
Linear Operator ◽  
Measure Space ◽  
Measure Zero ◽  
Positive Linear Operator ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Finite Measure Space ◽  
Image Position

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if(1.1)T is called sub-Markovian if(1.2)All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if(1.3)and(1.4)

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