Weighted ergodic theorems for weakly mixing transformation groups.

María Elena Becker

Displaying similar documents to “Weighted ergodic theorems for weakly mixing transformation groups.”

Weak almost periodicity of $L_{1}$ contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin (2000)

Studia Mathematica

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It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_{1}$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_{1}$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let...

On weakly mixing and doubly ergodic nonsingular actions

Sarah Iams, Brian Katz, Cesar E. Silva, Brian Street, Kirsten Wickelgren (2005)

Colloquium Mathematicae

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We study weak mixing and double ergodicity for nonsingular actions of locally compact Polish abelian groups. We show that if T is a nonsingular action of G, then T is weakly mixing if and only if for all cocompact subgroups A of G the action of T restricted to A is weakly mixing. We show that a doubly ergodic nonsingular action is weakly mixing and construct an infinite measure-preserving flow that is weakly mixing but not doubly ergodic. We also construct an infinite measure-preserving...

Conjugacies between ergodic transformations and their inverses

Geoffrey Goodson (2000)

Colloquium Mathematicae

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We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation $S T = T^{- 1} S$ . In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of $S^{2}$ . In particular, $S^{2}$ has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace $f \in L^{2} (X, ℱ, μ) : f (T^{2} x) = f (x)$ . For S and T ergodic satisfying this equation further constraints...

Bergelson's theorem for weakly mixing C*-dynamical systems

Rocco Duvenhage (2009)

Studia Mathematica

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We study a nonconventional ergodic average for asymptotically abelian weakly mixing C*-dynamical systems, related to a second iteration of Khinchin's recurrence theorem obtained by Bergelson in the measure-theoretic case. A noncommutative recurrence theorem for such systems is obtained as a corollary.

Counting and convergence in ergodic theory

Idris Assani, Zoltán Buczolich, Daniel R. Mauldin (2004)

Acta Universitatis Carolinae. Mathematica et Physica

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On new spectral multiplicities for ergodic maps

Alexandre I. Danilenko (2010)

Studia Mathematica

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It is shown that each subset of positive integers that contains 2 is realizable as the set of essential values of the multiplicity function for the Koopman operator of some weakly mixing transformation.

Power weakly mixing infinite transformations.

Day, Sarah L., Grivna, Brian R., McCartney, Earle P., Silva, Cesar E. (1999)

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Spectral properties of skew-product diffeomorphisms of tori

A. Iwanik (1997)

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On functions with scattered spectra on lea groups

Paweł Głowacki (1981)

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Infinite ergodic index $ℤ^{d}$ -actions in infinite measure

E. Muehlegger, A. Raich, C. Silva, M. Touloumtzis, B. Narasimhan, W. Zhao (1999)

Colloquium Mathematicae

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We construct infinite measure preserving and nonsingular rank one $ℤ^{d}$ -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving $ℤ^{d}$ -actions; for these we show that the individual basis transformations have conservative...

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

Studia Mathematica

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...