Displaying similar documents to “On some ergodic properties for continuous and affine functions”

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X i , i∈ I, and Y j , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of i I X i onto j J Y j . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of X i onto Y b ( i ) , i∈ I.

Self-affine measures and vector-valued representations

Qi-Rong Deng, Xing-Gang He, Ka-Sing Lau (2008)

Studia Mathematica

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Let A be a d × d integral expanding matrix and let S j ( x ) = A - 1 ( x + d j ) for some d j d , j = 1,...,m. The iterated function system (IFS) S j j = 1 m generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all f m -natural operators  transforming general affine connections on m -manifolds into general affine ones to the known description of all G L ( 𝐑 m ) -invariant maps 𝐑 m * 𝐑 m k 𝐑 m * k 𝐑 m for k = 1 , 3 .

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Kevin G. Hare, Nikita Sidorov (2015)

Studia Mathematica

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Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if | d e t M | 2 - 1 / d , then the attractor A M has non-empty interior. We also consider the set M of points in A M which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of M is positive. For this special class the full description of M is given as well. This paper continues our...

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 ) 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 , 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 , . . . , 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 d and the associated collection of rectangular parallelepipeds in d with sides parallel to the axes and dimensions of the form n × × n d with ( n , . . . , n d ) Γ . The associated multiparameter geometric and ergodic maximal operators M and M Γ are defined respectively on L ¹ ( d ) and L¹(Ω) by M g ( x ) = s u p x R 1 / | R | R | g ( y ) | d y and M Γ f ( ω ) = s u p ( n , . . . , n d ) Γ 1 / n n d j = 0 n - 1 j d = 0 n d - 1 | f ( U j 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 , . . . , T k , n ̅ = ( n , . . . , n k ) k and α ̅ = ( α , . . . , α k ) with 0 < α j 1 , we define the ergodic Cesàro-α̅ averages n ̅ , α ̅ f = 1 / ( j = 1 k A n j α j ) i k = 0 n k i = 0 n j = 1 k A n j - i j α j - 1 T k i k T i f . For these averages we prove the almost everywhere convergence on X and the convergence in the L p ( ν ) norm, when n , . . . , n k independently, for all f L p ( d ν ) with p > 1/α⁎ where α = m i n 1 j 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 ( 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 of integers such that l i m l | | | m l α | | | = 0 , there exists a continuous measure μ on the circle such that l i m l | | | 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 of integers such that | | | m l α | | | 0 and such that m l θ [ 1 ] is dense on the circle if and only if θ ∉ ℚα + ℚ.

Spectral radius of weighted composition operators in L p -spaces

Krzysztof Zajkowski (2010)

Studia Mathematica

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We prove that for the spectral radius of a weighted composition operator a T α , acting in the space L p ( X , , μ ) , the following variational principle holds: l n r ( a T α ) = m a x ν M ¹ α , e X l n | a | d ν , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, M ¹ α , e is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where = n = 0 α - n ( ) . This considerably extends the range of validity of the above formula, which was previously known...

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 ) ( τ x , φ ( x ) g ) and the cocycle ( φ ) n 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 ) m G ( d g ) or more generally of the τ φ -invariant measures μ χ ( d x ) χ ( g ) m G ( d g ) , where μ χ ( d x ) is χ∘φ-conformal for an exponential χ on G.

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ( Y ) ), and ϕ a linear isometry from M into C(Y) (resp., C ( Y ) ). We show, under the assumption that Π N Π T , where Π N is...

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let n = 1 a n be a convergent series of positive real numbers. L. Olivier proved that if the sequence ( a n ) is non-increasing, then lim n n a n = 0 . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n n a n = 0 ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence...

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

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We study the functional I f ( u ) = Ω f ( u ( x ) ) d x , where u=(u₁, ..., uₘ) and each u j is constant along some subspace W j of ℝⁿ. We show that if intersections of the W j ’s satisfy a certain condition then I f is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on W j j = 1 , . . . , m to have the equivalence: I f is weakly continuous if and only if f is Λ-affine.

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with s u p | | T / n w | | < for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) n - 1 k = 0 n - 1 T k converges uniformly; (ii) c l ( I - T ) X = z X : l i m n k = 1 n T k z / k e x i s t s .

Poincaré inequalities and rigidity for actions on Banach spaces

Piotr Nowak (2015)

Journal of the European Mathematical Society

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The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group G on a reflexive Banach space X has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that H 1 ( G , π ) = 0 for every isometric representation π of G on X . The condition is expressed in terms of p -Poincaré constants and we...

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ν ( X ) and ν ( X ) of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

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We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure C F L ( r ) of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space r of diameter r, is (isometrically if r = +∞) isomorphic to the space L I P ( r ) of equivalence classes of all real-valued Lipschitz maps on r . The space of all signed (real-valued) Borel measures on r is isometrically embedded in the dual space of C F L ( r ) and it is shown that the image...

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces X α , Y α and Z α such that (i) f X α , f Y α , f Z α = ω , where f is either trdef or ₀-trsur, (ii) A ( α ) - t r i n d X α = and M ( α ) - t r i n d X α = - 1 , (iii) A ( α ) - t r i n d Y α = - 1 and M ( α ) - t r i n d Y α = , and (iv) A ( α ) - t r i n d Z α = M ( α ) - t r i n d Z α = and A ( α + 1 ) M ( α + 1 ) - t r i n d Z α = - 1 . We also show that there exists no separable metrizable space W α with A ( α ) - t r i n d W α , M ( α ) - t r i n d W α and A ( α ) M ( α ) - t r i n d W α = , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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Equiintegrability in a compact interval E may be defined as a uniform integrability property that involves both the integrand f n and the corresponding primitive F n . The pointwise convergence of the integrands f n to some f and the equiintegrability of the functions f n together imply that f is also integrable with primitive F and that the primitives F n converge uniformly to F . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

Finiteness problems on Nash manifolds and Nash sets

José F. Fernando, José Manuel Gamboa, Jesús M. Ruiz (2014)

Journal of the European Mathematical Society

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We study here several finiteness problems concerning affine Nash manifolds M and Nash subsets X . Three main results are: (i) A Nash function on a semialgebraic subset Z of M has a Nash extension to an open semialgebraic neighborhood of Z in M , (ii) A Nash set X that has only normal crossings in M can be covered by finitely many open semialgebraic sets U equipped with Nash diffeomorphisms ( u 1 , , u m ) : U m such that U X = { u 1 u r = 0 } , (iii) Every affine Nash manifold with corners N is a closed subset of an affine Nash...

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let k be a field of characteristic p &gt; 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

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 ) and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n 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 { T H G } of compact subsets, there exists η > 0 such that # [ e ] Γ T = ( T ) + O ( ( T ) 1 - η ) for an explicit measure 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...

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...

On the duality between p -modulus and probability measures

Luigi Ambrosio, Simone Di Marino, Giuseppe Savaré (2015)

Journal of the European Mathematical Society

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Motivated by recent developments on calculus in metric measure spaces ( X , d , m ) , we prove a general duality principle between Fuglede’s notion [15] of p -modulus for families of finite Borel measures in ( X , d ) and probability measures with barycenter in L q ( X , m ) , with q dual exponent of p ( 1 , ) . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in X . In the final part of the paper we provide a new proof, independent of optimal transportation, of the...