-dimension function of bitopological spaces
M. Jelić (1987)
Matematički Vesnik
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M. Jelić (1987)
Matematički Vesnik
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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 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 , given by Urbański and Zinsmeister. The closure of the limit set of our IFS 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...
Yasunao Hattori, Jan van Mill (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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We prove that , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.
Anders Nilsson, Peter Wingren (2007)
Studia Mathematica
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A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have . Moreover, .
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 . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set is continuous at σ₀ as the function of the parameter if and only if . Since on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of on an open and dense subset of...
Tomasz Szarek, Maciej Ślęczka (2006)
Bulletin of the Polish Academy of Sciences. Mathematics
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It is shown that every Polish space X with admits a compact subspace Y such that where and denote the topological and Hausdorff dimensions, respectively.
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 . We prove that for any x ∈ (0,1), (x) contains a sequence increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure are nowhere dense.
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 with integer coefficients satisfying , i ≥ 1. In this case we say that 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...
Behrouz Taji (2014)
Annales de l’institut Fourier
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In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an provided that has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.
Chun-Yun Cao, Bao-Wei Wang, Jun Wu (2013)
Studia Mathematica
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Let be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence of integers, called the digit sequence of x, such that . We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set for any infinite subset B ⊂ ℕ, a question posed by...
Juan Rivera-Letelier (2001)
Fundamenta Mathematicae
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Given d ≥ 2 consider the family of polynomials for c ∈ ℂ. Denote by the Julia set of and let be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters : those for which the critical point 0 is not recurrent by and without parabolic cycles. The Hausdorff dimension of , denoted by , does not depend continuously on c at such ; on the other hand the function is analytic in . Our first result asserts that there is still some...
H. Murat Tuncali, Vesko Valov (2002)
Fundamenta Mathematicae
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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an -subset of X such that and the restriction is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...
Nicolas Monod, Henrik Densing Petersen (2014)
Annales de l’institut Fourier
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Let be any group containing an infinite elementary amenable subgroup and let . We construct an exhaustion of by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to -dimension and gives an answer to a question of Gaboriau.
Michael Levin (2003)
Fundamenta Mathematicae
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We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that we have and r is G-acyclic.
Thomas Gauthier (2012)
Annales scientifiques de l'École Normale Supérieure
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In the moduli space of degree rational maps, the bifurcation locus is the support of a closed positive current which is called the bifurcation current. This current gives rise to a measure whose support is the seat of strong bifurcations. Our main result says that has maximal Hausdorff dimension . As a consequence, the set of degree rational maps having distinct neutral cycles is dense in a set of full Hausdorff dimension.
Dylan Airey, Bill Mance, Joseph Vandehey (2016)
Czechoslovak Mathematical Journal
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Let be a sequence of bases with . In the case when the are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose -Cantor series expansion is both -normal and -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...
Dejun Wu (2015)
Czechoslovak Mathematical Journal
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We study the relations between finitistic dimensions and restricted injective dimensions. Let be a ring and a left -module with . If is selforthogonal, then we show that . Moreover, if is a left noetherian ring and is a finitely generated left -module with finite injective dimension, then . Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.
Kathryn E. Hare, Maria Roginskaya (2005)
Colloquium Mathematicae
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A measure is called -improving if it acts by convolution as a bounded operator from to for some q > p. Positive measures which are -improving are known to have positive Hausdorff dimension. We extend this result to complex -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 -functions.
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 with boundary and the spherical Hausdorff measure restricted to , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of is less than or equal to up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...
Yann Bugeaud, Arnaud Durand (2016)
Journal of the European Mathematical Society
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Let be a real number and let denote the set of real numbers approximable at order at least by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of is equal to . We investigate the size of the intersection of with Ahlfors regular compact subsets of the interval . In particular, we propose a conjecture for the exact value of the dimension of intersected with the middle-third Cantor set and give several...
Teo Mora (1989)
Publications mathématiques et informatique de Rennes
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John L. Lewis, Kaj Nyström, Andrew Vogel (2013)
Journal of the European Mathematical Society
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Let , and let , be given. In this paper we study the dimension of -harmonic measures that arise from non-negative solutions to the -Laplace equation, vanishing on a portion of , in the setting of -Reifenberg flat domains. We prove, for , that there exists small such that if is a -Reifenberg flat domain with , then -harmonic measure is concentrated on a set of -finite -measure. We prove, for , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of -harmonic...
Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)
Journal of the European Mathematical Society
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We take up the study of the Brill-Noether loci , where is a smooth projective variety of dimension , , and is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for , where is a divisor that moves linearly on a smooth projective variety of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension . In the -dimensional case...
Dachun Yang, Sibei Yang
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Let Φ be a concave function on (0,∞) of strictly critical lower type index and (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space via the local grand maximal function. Let for all t ∈ (0,∞). We also introduce the BMO-type space and establish the duality between and . Characterizations of , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...
Kuzman Adzievski (2006)
Annales Polonici Mathematici
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We study questions related to exceptional sets of pluri-Green potentials 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 are defined by , where for a fixed z ∈ B, denotes the holomorphic automorphism of B satisfying , and 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...