Higher order local dimensions and Baire category

Lars Olsen

Displaying similar documents to “Higher order local dimensions and Baire category”

Typical multifractal box dimensions of measures

L. Olsen (2011)

Fundamenta Mathematicae

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We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on $ℝ^{d}$ . We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_{α} : X \to Y (α < ω ₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $α < ω ₁ : f_{α} (x) \neq f (x)$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ ⁰_{η}$ sets which can be interesting in its own right.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β (f) \leq ω^{ξ ₁} \cdot ω^{ξ ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a $G_{δ}$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{< κ} = κ$ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense $G_{δ}$ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true $F_{σ δ}$ sets.

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].

Characterization of local dimension functions of subsets of $ℝ^{d}$

L. Olsen (2005)

Colloquium Mathematicae

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For a subset $E \subseteq ℝ^{d}$ and $x \in ℝ^{d}$ , the local Hausdorff dimension function of E at x is defined by $d i m_{H, l o c} (x, E) = l i m_{r ↘ 0} d i m_{H} (E \cap B (x, r))$ where $d i m_{H}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $κ^{κ}$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in $κ^{κ}$ for uncountable regular κ is however consistent (with...

Operators on the stopping time space

Dimitris Apatsidis (2015)

Studia Mathematica

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Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure $μ_{x * *}$ on the Cantor set. We use the measures $μ_{x * *} : x * * \in ℬ ₁ (S ¹)$ to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if $μ_{T * * (x * *)} : x * * \in ℬ ₁ (S ¹)$ is non-separable as a subset of $ℳ (2^{ℕ})$ .

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

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Let E be a Banach space and let $ℬ ₁ (B_{E *})$ and $₁ (B_{E *})$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E *}$ , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between $d i s t (f, ℬ ₁ (B_{E *}))$ and $d i s t (f, ₁ (B_{E *}))$ , where f is an affine function on $B_{E *}$ . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

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 $κ_{r} \in (0, \infty)$ , related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.

Baire one functions and their sets of discontinuity

Jonald P. Fenecios, Emmanuel A. Cabral, Abraham P. Racca (2016)

Mathematica Bohemica

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A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f : ℝ \to ℝ$ is of the first Baire class if and only if for each $ϵ > 0$ there is a sequence of closed sets ${C_{n}}_{n = 1}^{\infty}$ such that $D_{f} = ⋃_{n = 1}^{\infty} C_{n}$ and $ω_{f} (C_{n}) < ϵ$ for each $n$ where $ω_{f} (C_{n}) = sup {| f (x) - f (y) | : x, y \in C_{n}}$ and $D_{f}$ denotes the set of points of discontinuity of $f$ . The proof of the main theorem is based on a recent $ϵ$ - $δ$ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

Concerning Sets of the First Baire Category with Respect to Different Metrics

Maria Moszyńska, Grzegorz Sójka (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that if $ϱ_{H}$ and δ are the Hausdorff metric and the radial metric on the space ⁿ of star bodies in ℝ, with 0 in the kernel and with radial function positive and continuous, then a family ⊂ ⁿ that is meager with respect to $ϱ_{H}$ need not be meager with respect to δ. Further, we show that both the family of fractal star bodies and its complement are dense in ⁿ with respect to δ.

Transportation flow problems with Radon measure variables

Marcus Wagner (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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For a multidimensional control problem ${(P)}_{K}$ involving controls $u \in L_{\infty}$ , we construct a dual problem ${(D)}_{K}$ in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of $(L_{\infty}) *$ . For this purpose, we add to ${(P)}_{K}$ a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

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,...).

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...

Baire classes of complex $L_{1}$ -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

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Let $X$ be a complex $L_{1}$ -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- $α$ functions on the set $ext B_{X^{*}}$ of the extreme points of the dual unit ball $B_{X^{*}}$ to the whole unit ball $B_{X^{*}}$ . As a corollary we show that, given $α \in [1, ω_{1})$ , the intrinsic $α$ -th Baire class of $X$ can be identified with the space of bounded homogeneous Baire- $α$ functions on the set $ext B_{X^{*}}$ when $ext B_{X^{*}}$ satisfies certain topological assumptions. The paper is intended to be a complex counterpart to...

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

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara (2015)

Fundamenta Mathematicae

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Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$ . Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

Mean values and associated measures of $δ$ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let $u$ be a $δ$ -subharmonic function with associated measure $μ$ , and let $v$ be a superharmonic function with associated measure $ν$ , on an open set $E$ . For any closed ball $B (x, r)$ , of centre $x$ and radius $r$ , contained in $E$ , let $ℳ (u, x, r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s, t \to 0$ with $0 < s < t$ of the quotient $(ℳ (u, x, s) - ℳ (u, x, t)) / (ℳ (v, x, s) - ℳ (v, x, t))$ , lie between the upper and lower limits as $r \to 0 +$ of the quotient $μ (B (x, r)) / ν (B (x, r))$ . This enables us to use some well-known measure-theoretic results to prove new variants...

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 $x_{i} \in 0, 1$ . We prove that for any x ∈ (0,1), (x) contains a sequence ${β_{k}}_{k \geq 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\bar{(x)}$ are nowhere dense.

Epics in the category of $T_{2}$ k-groups need not have dense range

W. F. Lamartin (1976)

Colloquium Mathematicae

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Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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We show that, whenever $Γ$ is a countable abelian group and $Δ$ is a finitely-generated subgroup of $Γ$ , a generic measure-preserving action of $Δ$ on a standard atomless probability space $(X, μ)$ extends to a free measure-preserving action of $Γ$ on $(X, μ)$ . This extends a result of Ageev, corresponding to the case when $Δ$ is infinite cyclic.

Sets of $β$ -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

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We study natural measures on sets of $β$ -expansions and on slices through self similar sets. In the setting of $β$ -expansions, these allow us to better understand the measure of maximal entropy for the random $β$ -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...