Displaying similar documents to “Extensions of generic measure-preserving actions”

Approximation properties of β-expansions

Simon Baker (2015)

Acta Arithmetica

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Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ( ϵ i ) i = 1 0 , 1 a β-expansion for x if x = i = 1 ϵ i β - i . We call a finite sequence ( ϵ i ) i = 1 n 0 , 1 n an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given Ψ : 0 , we introduce the following subset of ℝ: W β ( Ψ ) : = m = 1 n = m ( ϵ i ) i = 1 n 0 , 1 n [ i = 1 n ( ϵ i ) / ( β i ) , i = 1 n ( ϵ i ) / ( β i ) + Ψ ( n ) ] In other words, W β ( Ψ ) is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities 0 x - i = 1 n ( ϵ i ) / ( β i ) Ψ ( n ) . When n = 1 2 n Ψ ( n ) < , the Borel-Cantelli lemma tells us that the Lebesgue measure...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space ( X , d , μ ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B ( x , r ) converge to f ( x ) when r converges to 0 . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

On strong measure zero subsets of κ 2

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

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We study the generalized Cantor space κ 2 and the generalized Baire space κ κ as analogues of the classical Cantor and Baire spaces. We equip κ κ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of κ 2 . We prove for successor κ = κ < κ that the ideal of strong measure zero sets of κ 2 is κ -additive, where κ is the size of the smallest unbounded family in κ κ , and that the generalized Borel...

Geometric rigidity of × m invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let μ be a probability measure on [ 0 , 1 ] which is invariant and ergodic for T a ( x ) = a x 𝚖𝚘𝚍 1 , and 0 < 𝚍𝚒𝚖 μ < 1 . Let f be a local diffeomorphism on some open set. We show that if E and ( f μ ) E μ E , then f ' ( x ) ± a r : r at μ -a.e. point x f - 1 E . In particular, if g is a piecewise-analytic map preserving μ then there is an open g -invariant set U containing supp μ such that g U is piecewise-linear with slopes which are rational powers of a . In a similar vein, for μ as above, if b is another integer and a , b are not powers of a common integer, and if ν is...

Convolution operators with anisotropically homogeneous measures on 2 n with n-dimensional support

E. Ferreyra, T. Godoy, M. Urciuolo (2002)

Colloquium Mathematicae

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Let α i , β i > 0 , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let t x = ( t α x , . . . , t α x ) , t x = ( t β x , . . . , t β x ) and | | x | | = i = 1 n | x i | 1 / α i . Let φ₁,...,φₙ be real functions in C ( - 0 ) such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on 2 n given by μ ( E ) = χ E ( x , φ ( x ) ) | | x | | γ - α d x , where α = i = 1 n α i and dx denotes the Lebesgue measure on ℝⁿ. Let T μ f = μ f and let | | T μ | | p , q be the operator norm of T μ from L p ( 2 n ) into L q ( 2 n ) , where the L p spaces are taken with respect to the Lebesgue measure. The type set E μ is defined by E μ = ( 1 / p , 1 / q ) : | | T μ | | p , q < , 1 p , q . In the case α i β k for 1 ≤ i,k ≤ n we characterize the...

Explicit computations of all finite index bimodules for a family of II 1 factors

Stefaan Vaes (2008)

Annales scientifiques de l'École Normale Supérieure

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We study II 1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index M - N -bimodule (in particular, every isomorphism between M and N ) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index M - M -bimodules is identified with an extended Hecke fusion algebra,...

L p -improving properties of certain singular measures on the Heisenberg group

Pablo Rocha (2022)

Mathematica Bohemica

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Let μ A be the singular measure on the Heisenberg group n supported on the graph of the quadratic function ϕ ( y ) = y t A y , where A is a 2 n × 2 n real symmetric matrix. If det ( 2 A ± J ) 0 , we prove that the operator of convolution by μ A on the right is bounded from L ( 2 n + 2 ) ( 2 n + 1 ) ( n ) to L 2 n + 2 ( n ) . We also study the type set of the measures d ν γ ( y , s ) = η ( y ) | y | - γ d μ A ( y , s ) , for 0 γ < 2 n , where η is a cut-off function around the origin on 2 n . Moreover, for γ = 0 we characterize the type set of ν 0 .

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q 1 restricted to E , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q 1 ( E ) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Radon-Nikodym property

Surjit Singh Khurana (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a Banach space E and a probability space ( X , 𝒜 , λ ) , a new proof is given that a measure μ : 𝒜 E , with μ λ , has RN derivative with respect to λ iff there is a compact or a weakly compact C E such that | μ | C : 𝒜 [ 0 , ] is a finite valued countably additive measure. Here we define | μ | C ( A ) = sup { k | μ ( A k ) , f k | } where { A k } is a finite disjoint collection of elements from 𝒜 , each contained in A , and { f k } E ' satisfies sup k | f k ( C ) | 1 . Then the result is extended to the case when E is a Frechet space.

On the characterization of harmonic functions with initial data in Morrey space

Bo Li, Jinxia Li, Bolin Ma, Tianjun Shen (2024)

Czechoslovak Mathematical Journal

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Let ( X , d , μ ) be a metric measure space satisfying the doubling condition and an L 2 -Poincaré inequality. Consider the nonnegative operator generalized by a Dirichlet form on X . We will show that a solution u to ( - t 2 + ) u = 0 on X × + satisfies an α -Carleson condition if and only if u can be represented as the Poisson integral of the operator with the trace in the generalized Morrey space L 2 , α ( X ) , where α is a nonnegative function defined on a class of balls in X . This result extends the analogous characterization...

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze, Michael Lin (2013)

Annales de l'I.H.P. Probabilités et statistiques

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It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n * f - f d m p 0 for every f L p , every (some) p [ 1 , ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f L p whenever p g t ; 1 . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of μ , for the strong sweeping out property (existence of a Borel set B with lim sup μ n * 1 B = 1 a.e. and lim inf μ n * 1 B = 0 a.e.). The results are extended to general compact Abelian groups...

Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

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For a smooth curve Γ and a set Λ in the plane 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia (2017)

Czechoslovak Mathematical Journal

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Let G be a finite group G , K a field of characteristic p 17 and let U be the group of units in K G . We show that if the derived length of U does not exceed 4 , then G must be abelian.

The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin, Lining Jiang, Zhenhua Ma (2015)

Czechoslovak Mathematical Journal

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Let G be a finite group and H a subgroup. Denote by D ( G ; H ) (or D ( G ) ) the crossed product of C ( G ) and H (or G ) with respect to the adjoint action of the latter on the former. Consider the algebra D ( G ) , e generated by D ( G ) and e , where we regard E as an idempotent operator e on D ( G ) for a certain conditional expectation E of D ( G ) onto D ( G ; H ) . Let us call D ( G ) , e the basic construction from the conditional expectation E : D ( G ) D ( G ; H ) . The paper constructs a crossed product algebra C ( G / H × G ) G , and proves that there is an algebra isomorphism between...