Bohr local properties of $C_Λ(T)$

Françoise Lust-Piquard

Displaying similar documents to “Bohr local properties of $C_{Λ} (T)$ ”

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_{G}$ . We consider the map $τ_{φ}$ defined on X × G by $τ_{φ} : (x, g) \mapsto (τ x, φ (x) g)$ and the cocycle ${(φ ₙ)}_{n \in ℤ}$ generated by φ. Using a characterization of the ergodic invariant measures for $τ_{φ}$ , we give the form of the ergodic decomposition of $μ (d x) \otimes m_{G} (d g)$ or more generally of the $τ_{φ}$ -invariant measures $μ_{χ} (d x) \otimes χ (g) m_{G} (d g)$ , where $μ_{χ} (d x)$ is χ∘φ-conformal for an exponential χ on G.

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. ...

Connectivity functions defined on $I^{n}$

Richard G. Gibson, Fred Roush (1987)

Colloquium Mathematicae

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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...

On the (C,α) Cesàro bounded operators

Elmouloudi Ed-dari (2004)

Studia Mathematica

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For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M ₙ^{α} = {(A ₙ^{α})}^{- 1} \sum_{k = 0}^{n} A_{n - k}^{α - 1} T^{k}$ . For α = 1, we find $M ₙ ¹ = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ , the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

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In ergodic theory, certain sequences of averages $A_{k} f$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence $A_{m_{k}} f$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_{k} f (x) = 1 / (2^{k}) \sum_{j = 4^{k} + 1}^{4^{k} + 2^{k}} f (T^{j} x)$ , then the subsequence $A_{k ²} f$ will not be pointwise good even on $L^{\infty}$ , but the subsequence $A_{2^{k}} f$ will be pointwise good on L¹. Understanding when the hyperexponential...

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, Alexander Stokolos (2012)

Fundamenta Mathematicae

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Let $U ₁, . . ., U_{d}$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $ℤ ₊^{d}$ and the associated collection of rectangular parallelepipeds in $ℝ^{d}$ with sides parallel to the axes and dimensions of the form $n ₁ \times \dots \times n_{d}$ with $(n ₁, . . ., n_{d}) \in Γ .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{Γ}$ are defined respectively on $L ¹ (ℝ^{d})$ and L¹(Ω) by $M_{} g (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | g (y) | d y$ and $M_{Γ} f (ω) = s u p_{(n ₁, . . ., n_{d}) \in Γ} 1 / n ₁ \dots n_{d} \sum_{j ₁ = 0}^{n ₁ - 1} \dots \sum_{j_{d} = 0}^{n_{d} - 1} | f (U ₁^{j ₁} \dots U_{d}^{j_{d}} ω) |$ . Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^{p} (ν)$ , $T ₁, . . ., T_{k}$ , $n ̅ = (n ₁, . . ., n_{k}) \in ℕ^{k}$ and $α ̅ = (α ₁, . . ., α_{k})$ with $0 < α_{j} \leq 1$ , we define the ergodic Cesàro-α̅ averages $_{n ̅, α ̅} f = 1 / (\prod_{j = 1}^{k} A_{n_{j}}^{α_{j}}) \sum_{i_{k} = 0}^{n_{k}} \dots \sum_{i ₁ = 0}^{n ₁} \prod_{j = 1}^{k} A_{n_{j} - i_{j}}^{α_{j} - 1} T_{k}^{i_{k}} \dots T ₁^{i ₁} f$ . For these averages we prove the almost everywhere convergence on X and the convergence in the $L^{p} (ν)$ norm, when $n ₁, . . ., n_{k} \to \infty$ independently, for all $f \in L^{p} (d ν)$ with p > 1/α⁎ where $α ⁎ = m i n_{1 \leq j \leq k} α_{j}$ . In the limit case p = 1/α⁎, we prove that the averages $_{n ̅, α ̅} f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $Λ (1 / α ⁎, φ_{m - 1})$ with $φ ₘ (t) = t {(1 + l o g ⁺ t)}^{m}$ . To obtain the result in the limit case we need...

On the convergence to 0 of mₙξmod 1

Bassam Fayad, Jean-Paul Thouvenot (2014)

Acta Arithmetica

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We show that for any irrational number α and a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $l i m_{l \to \infty} | | | m_{l} α | | | = 0$ , there exists a continuous measure μ on the circle such that $l i m_{l \to \infty} \int_{} | | | m_{l} θ | | | d μ (θ) = 0$ . This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $| | | m_{l} α | | | \to 0$ and such that $m_{l} θ [1]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

Distortion bounds for $C^{2 + η}$ unimodal maps

Mike Todd (2007)

Fundamenta Mathematicae

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We obtain estimates for derivative and cross-ratio distortion for $C^{2 + η}$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_{T} : T \to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...

La Queste del Saint $G R_{a}$ (AL): a Computational Approach to Local Algebra

Teo Mora (1989)

Publications mathématiques et informatique de Rennes

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Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $(x) = {g x + h_{j}}_{j = 1}^{k}$ of linear forms in ℤ[x], the set $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_{1}, . . ., δ_{m}$ form an increasing...

On connectivity properties of hyperspaces of $R_{0}$ spaces

M. Mršević (1979)

Matematički Vesnik

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Generalizations of Cesàro means and poles of the resolvent

Laura Burlando (2004)

Studia Mathematica

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An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that $n^{- 2} T ⁿ$ converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and $n^{- 3} \sum_{k = 0}^{n - 1} A^{k}$ converges uniformly, the sum of the range of 1-A and the kernel of (1-A)²...

Associated primes, integral closures and ideal topologies

Reza Naghipour (2006)

Colloquium Mathematicae

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Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set $A *_{a} (, N) : = ⋃_{n \geq 1} A s s_{R} R / {(ⁿ)}_{a}^{(N)}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${{(ⁿ)}_{a}^{(N)} :_{R} ⟨ ⟩}_{n \geq 1}$ is finer than the topology defined by ${(ⁿ)}_{a}^{(N)}_{n \geq 1}$ if and only if $A *_{a} (, N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover,...

Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

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Let $G : = SO {(n, 1)}^{\circ}$ and $Γ (n - 1) / 2$ for $n = 2, 3$ and when $δ > n - 2$ for $n \geq 4$ , we obtain an effective archimedean counting result for a discrete orbit of $Γ$ in a homogeneous space $H ∖ G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family ${ℬ_{T} \subset H ∖ G}$ of compact subsets, there exists $η > 0$ such that $# [e] Γ \cap ℬ_{T} = ℳ (ℬ_{T}) + O (ℳ {(ℬ_{T})}^{1 - η})$ for an explicit measure $ℳ$ on $H ∖ G$ which depends on $Γ$ . We also apply the affine sieve and describe the distribution of almost primes on orbits of $Γ$ in arithmetic...

On locally $S$ -closed spaces

Takashi Noiri (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano le condizioni sotto cui l’immagine (o l'immagine inversa) di uno spazio localmente $S$ -chiuso sia localmente $S$ -chiuso.

The one-sided ergodic Hilbert transform in Banach spaces

Guy Cohen, Christophe Cuny, Michael Lin (2010)

Studia Mathematica

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Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform $l i m_{n} \sum_{k = 1}^{n} (T^{k} x) / k$ . We prove that weak and strong convergence are equivalent, and in a reflexive space also $s u p_{n} | | \sum_{k = 1}^{n} (T^{k} x) / k | | < \infty$ is equivalent to the convergence. We also show that $- \sum_{k = 1}^{\infty} (T^{k}) / k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ${(I - T)}_{| \bar{(I - T) X}}^{r}$ .

Localization techniques in $L^{p}$ spaces

A. Pełczyński, H. Rosenthal (1975)

Studia Mathematica

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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...

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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Extremally disconnected resolutions of $T_{0}$ -spaces

A. Błaszczyk (1974)

Colloquium Mathematicae

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Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

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Let $f \in V_{r} () \cup_{r} ()$ , where, for 1 ≤ r < ∞, $V_{r} ()$ (resp., $_{r} ()$ ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...

The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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A further note on a class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Displaying similar documents to “Bohr local properties of C Λ ( T ) ”

Displaying similar documents to “Bohr local properties of $C_{Λ} (T)$ ”