Univoque sets for real numbers

Fan Lü; Bo Tan; Jun Wu

Displaying similar documents to “Univoque sets for real numbers”

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 $σ \in \bar{₀}$ if and only if $H D (J_{0, σ ₀}) \geq 4 / 3$ . 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...

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 $\partial E$ , 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 $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

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...

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials $P_{c} (z) = z^{d} + c$ for c ∈ ℂ. Denote by $J_{c}$ the Julia set of $P_{c}$ and let $ℳ_{d} = c | J_{c} i s c o n n e c t e d$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c ₀ \in \partial ℳ_{d}$ : those for which the critical point 0 is not recurrent by $P_{c ₀}$ and without parabolic cycles. The Hausdorff dimension of $J_{c}$ , denoted by $H D (J_{c})$ , does not depend continuously on c at such $c ₀ \in \partial ℳ_{d}$ ; on the other hand the function $c \mapsto H D (J_{c})$ is analytic in $ℂ - ℳ_{d}$ . Our first result asserts that there is still some...

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 $x = \sum_{i = 1}^{\infty} c_{i} q^{- i}$ with integer coefficients $c_{i}$ satisfying $0 \leq c_{i} < q$ , i ≥ 1. In this case we say that $(c_{i}) = c ₁ c ₂ . . .$ 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...

Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝔐$ and $ℜ$ be algebras of subsets of a set $Ω$ with $𝔐 \subset ℜ$ , and denote by $E (μ)$ the set of all quasi-measure extensions of a given quasi-measure $μ$ on $𝔐$ to $ℜ$ . We give some criteria for order boundedness of $E (μ)$ in $b a (ℜ)$ , in the general case as well as for atomic $μ$ . Order boundedness implies weak compactness of $E (μ)$ . We show that the converse implication holds under some assumptions on $𝔐$ , $ℜ$ and $μ$ or $μ$ alone, but not in general.

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

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Let $Q = {(q_{n})}_{n = 1}^{\infty}$ be a sequence of bases with $q_{i} \geq 2$ . In the case when the $q_{i}$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$ -Cantor series expansion is both $Q$ -normal and $Q$ -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$ , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

Kuzman Adzievski (2006)

Annales Polonici Mathematici

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We study questions related to exceptional sets of pluri-Green potentials $V_{μ}$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{μ}$ are defined by $V_{μ} (z) = \int_{B} l o g (1 / | ϕ_{z} (w) |) d μ (w)$ , where for a fixed z ∈ B, $ϕ_{z}$ denotes the holomorphic automorphism of B satisfying $ϕ_{z} (0) = z$ , $ϕ_{z} (z) = 0$ and $(ϕ_{z} \circ ϕ_{z}) (w) = w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

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A famous theorem of Carleson says that, given any function $f \in L^{p} (𝕋)$ , $p \in (1, + \infty)$ , its Fourier series $(S_{n} f (x))$ converges for almost every $x \in 𝕋$ . Beside this property, the series may diverge at some point, without exceeding $O (n^{1 / p})$ . We define the divergence index at $x$ as the infimum of the positive real numbers $β$ such that $S_{n} f (x) = O (n^{β})$ and we are interested in the size of the exceptional sets $E_{β}$ , namely the sets of $x \in 𝕋$ with divergence index equal to $β$ . We show that quasi-all functions in $L^{p} (𝕋)$ have a multifractal behavior with respect to...

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

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....

Norm continuity of pointwise quasi-continuous mappings

Alireza Kamel Mirmostafaee (2018)

Mathematica Bohemica

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Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $ϕ : X \to C_{p} (Y)$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y \times Y$ we will use topological games $𝒢_{1} (H)$ and $𝒢_{2} (H)$ on $Y \times Y$ between two players to prove that if the first player has a winning strategy in these games, then $ϕ$ is norm continuous on a dense $G_{δ}$ subset of $X$ . It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_{p} (Y)$ is norm continuous on a dense $G_{δ}$ subset of $X$ .

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 $Ω \subset ℝ^{n}, n \geq 3$ , and let $p, 1 < p < \infty, p \neq 2$ , 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 $\partial Ω$ , in the setting of $δ$ -Reifenberg flat domains. We prove, for $p \geq n$ , that there exists $\tilde{δ} = \tilde{δ} (p, n) > 0$ small such that if $Ω$ is a $δ$ -Reifenberg flat domain with $δ < \tilde{δ}$ , then $p$ -harmonic measure is concentrated on a set of $σ$ -finite $H^{n - 1}$ -measure. We prove, for $p \geq n$ , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$ -harmonic...

$R_{z}$ -supercontinuous functions

Davinder Singh, Brij Kishore Tyagi, Jeetendra Aggarwal, Jogendra K. Kohli (2015)

Mathematica Bohemica

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A new class of functions called “ $R_{z}$ -supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_{z}$ -supercontinuous functions properly includes the class of $R_{cl}$ -supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $cl$ -supercontinuous ( $\equiv$ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983),...