Typical multifractal box dimensions of measures

L. Olsen

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Displaying similar documents to “Typical multifractal box dimensions of measures”

The multifractal box dimensions of typical measures

Frédéric Bayart (2012)

Fundamenta Mathematicae

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We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of $ℝ^{d}$ . Our results are new even in the context of box dimensions of measures.

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...

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.

On the $k$ -Baire property

Alessandro Fedeli (1993)

Commentationes Mathematicae Universitatis Carolinae

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In this note we show the following theorem: “Let $X$ be an almost $k$ -discrete space, where $k$ is a regular cardinal. Then $X$ is $k^{+}$ -Baire iff it is a $k$ -Baire space and every point- $k$ open cover $𝒰$ of $X$ such that $card (𝒰) \leq k$ is locally- $k$ at a dense set of points.” For $k = ℵ_{0}$ we obtain a well-known characterization of Baire spaces. The case $k = ℵ_{1}$ is also discussed.

Higher order local dimensions and Baire category

Lars Olsen (2011)

Studia Mathematica

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Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension $d i m_{l o c} (μ; x)$ of a measure μ ∈ (X) at a point x ∈ X is defined by $d i m_{l o c} (μ; x) = l i m_{r ↘ 0} (l o g μ (B (x, r))) / (l o g r)$ whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local...

Insertion of a Contra-Baire- $1$ (Baire- $. 5$ ) Function

Majid Mirmiran (2019)

Communications in Mathematics

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Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- $. 5$ function between two comparable real-valued functions on the topological spaces that $F_{σ}$ -kernel of sets are $F_{σ}$ -sets.

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.

Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík, Jiří Spurný (2012)

Studia Mathematica

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If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball $B_{E *}$ that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of $B_{E *}$ , generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire...

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.

A model-theoretic Baire category theorem for simple theories and its applications

Ziv Shami (2013)

Fundamenta Mathematicae

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We prove a model-theoretic Baire category theorem for $τ ̃_{l o w}^{f}$ -sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in countable nfcp theories: either every type that is internal in a minimal type is essentially 1-based by means of the forking topologies, or T interprets an infinite definable 1-based group of finite D-rank or T interprets a strongly minimal formula.

Normal P-spaces and the $G_{δ}$ -topology

R. Levy, M. D. Rice (1981)

Colloquium Mathematicae

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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 $ℳ (2^{ℕ})$ .

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...

Generic power series on subsets of the unit disk

Balázs Maga, Péter Maga (2022)

Czechoslovak Mathematical Journal

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We examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $Λ$ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a nonreal boundary point on the unit circle, $f (U)$ is a dense set of $ℂ$ . As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property...

A remark on functions continuous on all lines

Luděk Zajíček (2019)

Commentationes Mathematicae Universitatis Carolinae

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We prove that each linearly continuous function $f$ on $ℝ^{n}$ (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for $f$ on an arbitrary Banach space $X$ , if $f$ has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such $f$ on a separable $X$ is continuous at all points outside a first category set which is also null in any usual...

On the Dirichlet problem for functions of the first Baire class

Jiří Spurný (2001)

Commentationes Mathematicae Universitatis Carolinae

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Let $ℋ$ be a simplicial function space on a metric compact space $X$ . Then the Choquet boundary $Ch X$ of $ℋ$ is an $F_{σ}$ -set if and only if given any bounded Baire-one function $f$ on $Ch X$ there is an $ℋ$ -affine bounded Baire-one function $h$ on $X$ such that $h = f$ on $Ch X$ . This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$ .