Ergodic transformations conjugate to their inverses by involutions

1996 ◽  
Vol 16 (1) ◽  
pp. 97-124 ◽  
Author(s):  
Geoffrey R. Goodson ◽  
Andrés del Junco ◽  
Mariusz Lemańczyk ◽  
Daniel J. Rudolph
Keyword(s):  
Discrete Spectrum ◽  
Probability Space ◽  
Closure Property ◽  
Simple Spectrum ◽  
Ergodic Transformations ◽  
Ergodic Automorphism ◽  
Singular Spectrum ◽  
Weak Closure ◽  
Identity Automorphism ◽  
Borel Probability

AbstractLetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

Orbit equivalence and actions of

2006 ◽  
Vol 71 (1) ◽  
pp. 265-282 ◽  
Author(s):  
Asge Törnquist
Keyword(s):  
Probability Space ◽  
Free Groups ◽  
Orbit Equivalence ◽  
Borel Probability ◽  
Free Actions ◽  
Measure Preserving Transformations

AbstractIn this paper we show that there are “E0 many” orbit inequivalent free actions of the free groups , 2 ≤ n ≤ ∞ by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.

Poorly Approximated ℤ2-Cocycles For Transformations With Rational Discrete Spectrum

1991 ◽  
Vol 34 (3) ◽  
pp. 338-342
Author(s):  
Adam Fieldsteel
Keyword(s):  
Discrete Spectrum ◽  
The Other ◽  
Ergodic Automorphism ◽  
Point Extension ◽  
Other Hand

AbstractLet T be an ergodic automorphism with rational discrete spectrum and ϕ a ℤ2-cocyle for T. We show that the resulting two-point extension of T is cohomologous to a Morse cocycle if ϕ is approximated with speed o(1/n).On the other hand, we show by example that this is in general false when the speed of approximation is O(1/n).

REPRESENTATION OF PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC BURGERS' EQUATIONS

2009 ◽  
Vol 09 (04) ◽  
pp. 613-634 ◽  
Author(s):  
YONG LIU ◽  
HUAIZHONG ZHAO
Keyword(s):  
Dynamical System ◽  
Integral Equation ◽  
Brownian Motion ◽  
Linear Operator ◽  
Probability Space ◽  
Burgers Equation ◽  
Stationary Solutions ◽  
Random Dynamical System ◽  
Ergodic Transformations ◽  
Stochastic Burgers Equation

In this paper, we show that the stationary solution u(t, ω) of the differentiable random dynamical system U: ℝ+ × L2[0, 1] × Ω → L2[0, 1] generated by the stochastic Burgers' equation with large viscosity, denoted by ν, driven by a Brownian motion in L2[0, 1], is given by: u(t, ω) = U(t, Y(ω), ω) = Y(θ(t, ω)), where Y(ω) can be represented by the following integral equation: [Formula: see text] Here θ is the group of P-preserving ergodic transformations on the canonical probability space [Formula: see text] such that θ(t, ω)(s) = W(t + s) - W(t), where W is the L2[0, 1]-valued Brownian motion on the probability space [Formula: see text], Tν is the linear operator semigroup on L2[0, 1] generated by νΔ.

scholarly journals Pointwise equidistribution and translates of measures on homogeneous spaces

2018 ◽  
Vol 40 (2) ◽  
pp. 453-477 ◽  
Author(s):  
OSAMA KHALIL
Keyword(s):  
Probability Measure ◽  
Lie Groups ◽  
Probability Space ◽  
Homogeneous Spaces ◽  
Horocycle Flow ◽  
Closed Orbits ◽  
Upper Density ◽  
Borel Probability ◽  
General Conditions

Let $(X,\mathfrak{B},\unicode[STIX]{x1D707})$ be a Borel probability space. Let $T_{n}:X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\unicode[STIX]{x1D708}$ be a probability measure on $X$ such that $(1/N)\sum _{n=1}^{N}(T_{n})_{\ast }\unicode[STIX]{x1D708}\rightarrow \unicode[STIX]{x1D707}$ in the weak-$\ast$ topology. Under general conditions, we show that for $\unicode[STIX]{x1D708}$ almost every $x\in X$, the measures $(1/N)\sum _{n=1}^{N}\unicode[STIX]{x1D6FF}_{T_{n}x}$ become equidistributed towards $\unicode[STIX]{x1D707}$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prove equidistribution of exponentially sparse orbits of the horocycle flow on quotients of $\text{SL}(2,\mathbb{R})$, starting from every point in almost every direction.

The Group Aut (μ) is Roelcke Precompact

2012 ◽  
Vol 55 (2) ◽  
pp. 297-302 ◽  
Author(s):  
Eli Glasner
Keyword(s):  
Probability Space ◽  
Separable Hilbert Space ◽  
Continuous Functions ◽  
Uniform Structure ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Markov Operators ◽  
Weakly Almost Periodic ◽  
Borel Probability ◽  
Uniformly Continuous

AbstractFollowing a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group G = Aut(μ) of automorphisms of an atomless standard Borel probability space (X, μ) is precompact. We identify the corresponding compactification as the space of Markov operators on L2(μ) and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on G, i.e., functions on G arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that G is totally minimal.

Borel equivalence relations which are highly unfree

2008 ◽  
Vol 73 (4) ◽  
pp. 1271-1277 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Probability Space ◽  
Ergodic Measure ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Borel Probability

AbstractThere is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that E∣c is not essentially free on any conull C ⊂ X.

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.

scholarly journals Transformations With Discrete Spectrum are Stacking Transformations

1976 ◽  
Vol 28 (4) ◽  
pp. 836-839 ◽  
Author(s):  
Andrés Del Junco
Keyword(s):  
Discrete Spectrum ◽  
Ergodic Theory ◽  
Great Success ◽  
Ergodic Transformations ◽  
Simple Characterization

The stacking method (see [1] and [5, Section 6]) has been used with great success in ergodic theory to construct a wide variety of examples of ergodic transformations (see, for example, [1 ; 3 ; 4; 5; 7]). However very little is known in general about the class of transformations which can be constructed by the stacking method using single stacks. In particular there is no simple characterization of the class .

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