Metastability in the Furstenberg-Zimmer tower

Jeremy Avigad; Henry Towsner

Displaying similar documents to “Metastability in the Furstenberg-Zimmer tower”

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.

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

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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...

Dispersing cocycles and mixing flows under functions

Klaus Schmidt (2002)

Fundamenta Mathematicae

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Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X,,μ) and A a locally compact second countable abelian group. A cocycle c: G × X → A for T disperses if $l i m_{g \to \infty} c (g, \cdot) - α (g) = \infty$ in measure for every map α: G → A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A₀ ⊂ A such that the composition θ ∘ c: G × X → A/A₀ of c with the quotient map θ: A → A/A₀ is trivial (i.e. cohomologous to a homomorphism η: G → A/A₀). This...

Invariance of the Gibbs measure for the Benjamin–Ono equation

Yu Deng (2015)

Journal of the European Mathematical Society

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In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than $L^{2}$ , extending the $L^{2}$ well-posedness result of Molinet [20].

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.

Distinguished subspaces of $L_{p}$ of maximal dimension

Geraldo Botelho, Daniel Cariello, Vinícius V. Fávaro, Daniel Pellegrino, Juan B. Seoane-Sepúlveda (2013)

Studia Mathematica

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Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

Studia Mathematica

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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...

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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A note on the strong maximal operator on ℝⁿ

Jiecheng Chen, Xiangrong Zhu (2004)

Studia Mathematica

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We prove that for f ∈ L ln⁺L(ℝⁿ) with compact support, there is a g ∈ L ln⁺L(ℝⁿ) such that (a) g and f are equidistributed, (b) $M_{S} (g) \in L ¹ (E)$ for any measurable set E of finite measure.

Large sets of integers and hierarchy of mixing properties of measure preserving systems

Vitaly Bergelson, Tomasz Downarowicz (2008)

Colloquium Mathematicae

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We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R_{A, B}^{ε} = n \in ℤ : μ (A \cap T ⁿ B) > μ (A) μ (B) - ε$ of times of “fat intersection”. Among other things we show that the sets $R_{A, B}^{ε}$ allow one...

On the maximal function for rotation invariant measures in $ℝ^{n}$

Ana Vargas (1994)

Studia Mathematica

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Given a positive measure μ in $ℝ^{n}$ , there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{μ} f (x) = s u p_{x \in B} 1 / μ (B) ʃ_{B} | f | d μ$ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in $ℝ^{n}$ . We give some necessary and sufficient conditions for $M_{μ}$ to be bounded from $L^{1} (d μ)$ to $L^{1, \infty} (d μ)$ .

On the Rademacher maximal function

Mikko Kemppainen (2011)

Studia Mathematica

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This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^{p}$ -boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^{p}$ -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale...

Template iterations and maximal cofinitary groups

Vera Fischer, Asger Törnquist (2015)

Fundamenta Mathematicae

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Jörg Brendle (2003) used Hechler’s forcing notion for adding a maximal almost disjoint family along an appropriate template forcing construction to show that (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that $_{g}$ , the minimal size of a maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys...

A ratio ergodic theorem for multiparameter non-singular actions

Michael Hochman (2010)

Journal of the European Mathematical Society

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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...

Solvability of the functional equation f = (T-I)h for vector-valued functions

Ryotaro Sato (2004)

Colloquium Mathematicae

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Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $l i m_{n \to \infty} f ₙ = f$ in measure for some f ∈ M(μ;X), then also $l i m_{n \to \infty} T f ₙ = T f$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X)...

Problems on averages and lacunary maximal functions

Andreas Seeger, James Wright (2011)

Banach Center Publications

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We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1, \infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^{p}$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^{p}$ ...

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space $L^{p} (μ, X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of $L^{p} (μ, X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

On invariant measures for piecewise $C^{2}$ -transformations of the n-dimensional cube

M. Jabłoński (1983)

Annales Polonici Mathematici

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Measure preserving analytic diffeomorphisms of countable dense sets in $C^{n}$ and $R^{n}$

Michał Morayne (1987)

Colloquium Mathematicae

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Non-Typical Points for β-Shifts

David Färm, Tomas Persson (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study sets of non-typical points under the map $f_{β} \mapsto β x$ mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β > 1.541 found in the above paper can be removed.

Spaces with maximal projection constants

Hermann König, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

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We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{\infty}$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.

A note on rare maximal functions

Paul Alton Hagelstein (2003)

Colloquium Mathematicae

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A necessary and sufficient condition is given on the basis of a rare maximal function $M_{l}$ such that $M_{l} f \in L ¹ ([0, 1])$ implies f ∈ L log L([0,1]).

Unique Bernoulli $g$ -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

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We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$ -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$ -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$ -measure.

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x \in ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ} (μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty}^{ϱ} (μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

The maximal theorem for weighted grand Lebesgue spaces

Alberto Fiorenza, Babita Gupta, Pankaj Jain (2008)

Studia Mathematica

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We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${| | M f | |}_{p), w} {\leq c | | f | |}_{p), w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$ , and therefore iff ${| | M f | |}_{p, w} {\leq c | | f | |}_{p, w}$ for some c independent of f. Some results of similar type are discussed for the case of small...

An anti-classification theorem for ergodic measure preserving transformations

Matthew Foreman, Benjamin Weiss (2004)

Journal of the European Mathematical Society

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Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT’s on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos–von Neumann theorem classifying ergodic MPT’s with pure point spectrum. We treat both the classical...

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

H¹ and BMO for certain locally doubling metric measure spaces of finite measure

Andrea Carbonaro, Giancarlo Mauceri, Stefano Meda (2010)

Colloquium Mathematicae

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In a previous paper the authors developed an H¹-BMO theory for unbounded metric measure spaces (M,ρ,μ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric” property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies...

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.

Topology and measure of buried points in Julia sets

Clinton P. Curry, John C. Mayer, E. D. Tymchatyn (2013)

Fundamenta Mathematicae

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It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense $G_{δ}$ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points....

Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of $L^{p} (μ)$ , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $s u p_{n \in ℤ} | | U ⁿ | | < \infty$ . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n \geq 1} (U ⁿ f) / n^{1 - α}$ , 0 ≤ α < 1, in terms of $| | f + \dots + U^{n - 1} f {| |}_{p}$ , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)

Fundamenta Mathematicae

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Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ${ϕ |}_{X ₀}$ is continuous in the relative topology on X₀ and $ϕ^{- 1} |_{Y ₀}$ is continuous in the relative topology on Y₀, (2) $ϕ (O r b_{T} (x)) = O r b_{S} (ϕ (x))$ for μ-a.e. x ∈ X. (X,,μ,T)...

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

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We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p, q}$ , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p, q, r}$ such that for any $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

Example of a mean ergodic L¹ operator with the linear rate of growth

Wojciech Kosek (2011)

Colloquium Mathematicae

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The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty$ . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $l i m s u p | | T ⁿ | | / n^{1 - γ ₀} = 0 .$ In this note we construct an example of a nonpositive L¹ operator with the...

Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

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Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$ . A loop $h : S^{1} \to G$ is called strictly ergodic if for some irrational number the associated skew product map $T : S^{1} \times Y \to S^{1} \times Y$ defined by $T (t, y) = (t + α; h (t) y)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply...

On Peano derivatives in $L^{p} (E_{n})$

S. Lasher (1968)

Studia Mathematica

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Less than $2^{ω}$ many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

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We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $c o f () < 2^{ω}$ ) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ.

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.