An anti-classification theorem for ergodic measure preserving transformations

Matthew Foreman; Benjamin Weiss

Displaying similar documents to “An anti-classification theorem for ergodic measure preserving transformations”

Ergodicity and conservativity of products of infinite transformations and their inverses

Julien Clancy, Rina Friedberg, Indraneel Kasmalkar, Isaac Loh, Tudor Pădurariu, Cesar E. Silva, Sahana Vasudevan (2016)

Colloquium Mathematicae

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We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T is ergodic, but the product $T \times T^{- 1}$ is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

On measure theoretical analogues of the Takesaki structure theorem for type III factors

Alexandre Danilenko, Toshihiro Hamachi (2000)

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The orbit equivalence of type $I I I_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $I I I_{0}$ cocycles with values in Abelian groups.

Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

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Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$ -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$ -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these...

Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends, Jörg Schmeling (2001)

Czechoslovak Mathematical Journal

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We call a sequence $(T_{n})$ of measure preserving transformations strongly mixing if $P (T_{n}^{- 1} A \cap B)$ tends to $P (A) P (B)$ for arbitrary measurable $A$ , $B$ . We investigate whether one can pass to a suitable subsequence $(T_{n_{k}})$ such that $\frac{1}{K} \sum_{k = 1}^{K} f (T_{n_{k}}) ⟶ \int f d P$ almost surely for all (or “many”) integrable $f$ .

A ratio ergodic theorem for multiparameter non-singular actions

Michael Hochman (2010)

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We prove a ratio ergodic theorem for non-singular free $ℤ^{d}$ and $ℝ^{d}$ actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in $ℝ^{d}$ . The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general...

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...

Characterizing mildly mixing actions by orbit equivalence of products.

Hawkins, Jane, Silva Cesar, E. (1998)

The New York Journal of Mathematics [electronic only]

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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...

JOP's counting function and Jones' square function

Karin Reinhold (2006)

Studia Mathematica

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We study a class of square functions in a general framework with applications to a variety of situations: samples along subsequences, averages of $ℤ ₊^{d}$ actions and of positive L¹ contractions. We also study the relationship between a counting function first introduced by Jamison, Orey and Pruitt, in a variety of situations, and the corresponding ergodic averages. We show that the maximal counting function is not dominated by the square functions.

Infinite measure preserving flows with infinite ergodic index

Alexandre I. Danilenko, Anton V. Solomko (2009)

Colloquium Mathematicae

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We construct a rank-one infinite measure preserving flow ${(T_{r})}_{r \in ℝ}$ such that for each p > 0, the “diagonal” flow ${(T_{r} \times \dots \times T_{r})}_{r \in ℝ} (p t i m e s)$ on the product space is ergodic.

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...

Predictability, entropy and information of infinite transformations

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We show that a certain type of quasifinite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasifinite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with normalization ∝ √n. Lastly, we show that certain ergodic, probability...

Ergodic transforms associated to general averages

H. Aimar, A. L. Bernardis, F. J. Martín-Reyes (2010)

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Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^{p}$ , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^{p}$ . For p = 1 we find that the...

Examples of minimal diffeomorphisms on 𝕋² semiconjugate to an ergodic translation

Alejandro Passeggi, Martín Sambarino (2013)

Fundamenta Mathematicae

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We prove that for every ϵ > 0 there exists a minimal diffeomorphism f: ² → ² of class $C^{3 - ϵ}$ and semiconjugate to an ergodic translation with the following properties: zero entropy, sensitivity to initial conditions, and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé’s example of a derived-from-Anosov diffeomorphism on ³.

An algebraic group associated to an ergodic diffeomorphism

Robert J. Zimmer (1981)

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Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

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In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any...

Ergodic decomposition of entropy

S. Janković (1989)

Matematički Vesnik

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Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_{k}$ such that the weighted ergodic averages corresponding to $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...