Displaying similar documents to “Dynamics of finite-multivalued transformations.”

Ergodic seminorms for commuting transformations and applications

Bernard Host (2009)

Studia Mathematica

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Recently, T. Tao gave a finitary proof of a convergence theorem for multiple averages with several commuting transformations, and soon thereafter T. Austin gave an ergodic proof of the same result. Although we give here another proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they may...

Weakly mixing transformations and the Carathéodory definition of measurable sets

Amos Koeller, Rodney Nillsen, Graham Williams (2007)

Colloquium Mathematicae

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Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 𝕋 given by T(z) = vz. It is known that the problem of calculating the outer measure of a T-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, T is not weakly mixing. Now there is an example, due to Kakutani, of a transformation ψ̃ which is weakly mixing but...

On a pointwise ergodic theorem for multiparameter semigroups.

Ryotaro Sato (1994)

Publicacions Matemàtiques

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Let Ti (i = 1, 2, ..., d) be commuting null preserving transformations on a finite measure space (X, F, μ) and let 1 ≤ p < ∞. In this paper we prove that for every f ∈ Lp(μ) the averages Anf(x) = (n + 1)-d Σ0≤ni≤n f(T1 n1 T2 n2...

Weak mixing of a transformation similar to Pascal

Daniel M. Kane (2007)

Colloquium Mathematicae

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We construct a class of transformations similar to the Pascal transformation, except for the use of spacers, and show that these transformations are weakly mixing.

Tower multiplexing and slow weak mixing

Terrence Adams (2015)

Colloquium Mathematicae

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A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity sequences of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical...

Some thoughts about Segal's ergodic theorem

Daniel W. Stroock (2010)

Colloquium Mathematicae

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Over fifty years ago, Irving Segal proved a theorem which leads to a characterization of those orthogonal transformations on a Hilbert space which induce ergodic transformations. Because Segal did not present his result in a way which made it readily accessible to specialists in ergodic theory, it was difficult for them to appreciate what he had done. The purpose of this note is to state and prove Segal's result in a way which, I hope, will win it the recognition which it deserves. ...

Mixing on rank-one transformations

Darren Creutz, Cesar E. Silva (2010)

Studia Mathematica

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We prove that mixing on rank-one transformations is equivalent to "the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums". In particular, all polynomial staircase transformations are mixing.

On v-positive type transformations in infinite measure

Tudor Pădurariu, Cesar E. Silva, Evangelie Zachos (2015)

Colloquium Mathematicae

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For each vector v we define the notion of a v-positive type for infinite-measure-preserving transformations, a refinement of positive type as introduced by Hajian and Kakutani. We prove that a positive type transformation need not be (1,2)-positive type. We study this notion in the context of Markov shifts and multiple recurrence, and give several examples.

A joint limit theorem for compactly regenerative ergodic transformations

David Kocheim, Roland Zweimüller (2011)

Studia Mathematica

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We study conservative ergodic infinite measure preserving transformations satisfying a compact regeneration property introduced by the second-named author in J. Anal. Math. 103 (2007). Assuming regular variation of the wandering rate, we clarify the asymptotic distributional behaviour of the random vector (Zₙ,Sₙ), where Zₙ and Sₙ are respectively the time of the last visit before time n to, and the occupation time of, a suitable set Y of finite measure.

Ergodic properties of a class of discrete Abelian group extensions of rank-one transformations

Chris Dodd, Phakawa Jeasakul, Anne Jirapattanakul, Daniel M. Kane, Becky Robinson, Noah D. Stein, Cesar E. Silva (2010)

Colloquium Mathematicae

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We define a class of discrete Abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that Cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.