Displaying similar documents to “Sets of β -expansions and the Hausdorff measure of slices through fractals”

L p -improving properties of measures of positive energy dimension

Kathryn E. Hare, Maria Roginskaya (2005)

Colloquium Mathematicae

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A measure is called L p -improving if it acts by convolution as a bounded operator from L p to L q for some q > p. Positive measures which are L p -improving are known to have positive Hausdorff dimension. We extend this result to complex L p -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of L p -functions.

Measures of maximal entropy for random β -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

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Let β > 1 be a non-integer. We consider β -expansions of the form i = 1 d i / β i , where the digits ( d i ) i 1 are generated by means of a Borel map K β defined on { 0 , 1 } × [ 0 , β / ( β 1 ) ] . We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits ( d i ) i 1 form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness...

On Ordinary and Standard Lebesgue Measures on

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on 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 for α = (1,1,...).

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality u ( m ) w ( u ) . Next, we apply this result to obtain existence theorems for the equation x ( m ) = f ( t , x ) .

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.

Invariant densities for random β -expansions

Karma Dajani, Martijn de Vries (2007)

Journal of the European Mathematical Society

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Let β > 1 be a non-integer. We consider expansions of the form i = 1 d i / β i , where the digits ( d i ) i 1 are generated by means of a Borel map K β defined on { 0 , 1 } × [ 0 , β ( β 1 ) ] . We show existence and uniqueness of a K β -invariant probability measure, absolutely continuous with respect to m p λ , where m p is the Bernoulli measure on { 0 , 1 } with parameter p ( 0 < p < 1 ) and λ is the normalized Lebesgue measure on [ 0 , β ( β 1 ) ] . Furthermore, this measure is of the form m p μ β , p , where μ β , p is equivalent to λ . We prove that the measure of maximal entropy and m p λ are mutually...

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube I M = [ 0 , 1 ) M , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on I M ; (3) the measures have the convolution property that μ L p L p + ε for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then μ L p L q for any measure μ in our...

On inhomogeneous self-similar measures and their L q spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

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Let S i : d d for i = 1,..., N be contracting similarities, let ( p , . . . , p N , p ) be a probability vector and let ν be a probability measure on d with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on d such that μ = i = 1 N p i μ S i - 1 + p ν . We give satisfactory estimates for the lower and upper bounds of the L q spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

Univoque sets for real numbers

Fan Lü, Bo Tan, Jun Wu (2014)

Fundamenta Mathematicae

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For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with x i 0 , 1 . We prove that for any x ∈ (0,1), (x) contains a sequence β k k 1 increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure ( x ) ¯ are nowhere dense.

Quantization Dimension Estimate of Inhomogeneous Self-Similar Measures

Mrinal Kanti Roychowdhury (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider an inhomogeneous measure μ with the inhomogeneous part a self-similar measure ν, and show that for a given r ∈ (0,∞) the lower and the upper quantization dimensions of order r of μ are bounded below by the quantization dimension D r ( ν ) of ν and bounded above by a unique number κ r ( 0 , ) , related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.

L p - L q estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy, P. Rocha (2013)

Colloquium Mathematicae

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We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by ν ( E ) = χ E ( w , φ ( w ) ) η ( w ) d w , where φ ( w ) = j = 1 n a j | w j | ² , w = (w₁,...,wₙ) ∈ ℂⁿ, a j , and η(w) = η₀(|w|²) with η C c ( ) . We characterize the set of pairs (p,q) such that the convolution operator with ν is L p ( ) - L q ( ) bounded. We also obtain L p -improving properties of measures supported on the graph of the function φ ( w ) = | w | 2 m .

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,...

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from L p ( ) to L q ( ) . We also give a condition on p which is necessary if this operator maps L p ( ) into L²().

The type set for homogeneous singular measures on ℝ ³ of polynomial type

E. Ferreyra, T. Godoy (2006)

Colloquium Mathematicae

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Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by μ ( E ) = D χ E ( x , φ ( x ) ) d x with D = x ∈ ℝ ²:|x| ≤ 1 and let T μ be the convolution operator with the measure μ. Let φ = φ e φ e be the decomposition of φ into irreducible factors. We show that if e i m / 2 for each φ i of degree 1, then the type set E μ : = ( 1 / p , 1 / q ) [ 0 , 1 ] × [ 0 , 1 ] : | | T μ | | p , q < can be explicitly described as a closed polygonal region.

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...

A two-dimensional univoque set

Martijn de Vrie, Vilmos Komornik (2011)

Fundamenta Mathematicae

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Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form x = i = 1 c i q - i with integer coefficients c i satisfying 0 c i < q , i ≥ 1. In this case we say that ( c i ) = c c . . . is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also...

Asymptotic nature of higher Mahler measure

(2014)

Acta Arithmetica

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We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure m k ( P ) of a polynomial P , where m k ( P ) is the integral of l o g k | P | over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding m k ( P ) , in particular | m k ( P ) | / k ! 1 / π as k → ∞.

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets J 0 , σ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of J 0 , σ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS ϕ σ , α n , k is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure μ S : = a S ( 1 - | a | ² ) δ a is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure μ S bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Self-affine measures that are L p -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called L p -improving if it acts by convolution as a bounded operator from L q to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are L p -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be L p -improving.

On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas (2011)

Fundamenta Mathematicae

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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map g σ . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set J 0 , σ is continuous at σ₀ as the function of the parameter σ ¯ if and only if H D ( J 0 , σ ) 4 / 3 . Since H D ( J 0 , σ ) > 4 / 3 on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of H D ( J 0 , σ ) on an open and dense subset of...

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

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Let ( X , d X ) , ( Ω , d Ω ) be complete separable metric spaces. Denote by (X) the space of probability measures on X, by W p the p-Wasserstein metric with some p ∈ [1,∞), and by p ( X ) the space of probability measures on X with finite Wasserstein distance from any point measure. Let f : Ω p ( X ) , ω f ω , be a Borel map such that f is a contraction from ( Ω , d Ω ) into ( p ( X ) , W p ) . Let ν₁,ν₂ be probability measures on Ω with W p ( ν , ν ) finite. On X we consider the subordinated measures μ i = Ω f ω d ν i ( ω ) . Then W p ( μ , μ ) W p ( ν , ν ) . As an application we show that the solution measures ϱ α ( t ) ...

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 nearly radial marginals of high-dimensional probability measures

Bo&#039;az Klartag (2010)

Journal of the European Mathematical Society

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Suppose that μ is an absolutely continuous probability measure on R n, for large n . Then μ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ( C / ε ) C d , then there exist d -dimensional marginals of μ that are ε -far from being sphericallysymmetric, in an appropriate sense. Here C > 0 is a universal constant.

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...

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

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We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in n i , i ≤ m, then for every F in the Grassmannian G N , n , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, π F ( μ μ ) , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

On biorthogonal systems whose functionals are finitely supported

Christina Brech, Piotr Koszmider (2011)

Fundamenta Mathematicae

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We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space K 2 n such that C ( K 2 n ) has no uncountable (semi)biorthogonal sequence ( f ξ , μ ξ ) ξ ω where μ ξ ’s are atomic measures with supports consisting of at most 2n-1 points of K 2 n , but has biorthogonal systems ( f ξ , μ ξ ) ξ ω where μ ξ ’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace 𝒦 = 𝒦 T ( B ) affiliated with ℳ, such that the Brown measure of T | 𝒦 is concentrated...

On the dimension of p -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

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Let Ω n , n 3 , and let p , 1 < p < , p 2 , be given. In this paper we study the dimension of p -harmonic measures that arise from non-negative solutions to the p -Laplace equation, vanishing on a portion of Ω , in the setting of δ -Reifenberg flat domains. We prove, for p n , that there exists δ ˜ = δ ˜ ( p , n ) > 0 small such that if Ω is a δ -Reifenberg flat domain with δ < δ ˜ , then p -harmonic measure is concentrated on a set of σ -finite H n 1 -measure. We prove, for p n , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p -harmonic...

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...

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 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...