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Displaying similar documents to “ ℰ B -dimension function of bitopological spaces”

On the Separation Dimension of K ω

Yasunao Hattori, Jan van Mill (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that t r t K ω > ω + 1 , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

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 ∈ ₀ ¯ if and only if H D ( J 0 , σ ₀ ) ≥ 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...

An obstruction to ℓ p -dimension

Nicolas Monod, Henrik Densing Petersen (2014)

Annales de l’institut Fourier

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Let G be any group containing an infinite elementary amenable subgroup and let 2 < p < ∞ . We construct an exhaustion of ℓ p G by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to ℓ p -dimension and gives an answer to a question of Gaboriau.

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...

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.

On C * -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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On dimensionally restricted maps

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 F σ -subset A k of X such that d i m A k ≤ k and the restriction f | ( X ∖ A k ) 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...