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1
Content available remote McMullen’s and geometric pressures and approximating the Hausdorff dimension of Julia sets from below
Przytycki Feliks
Bulletin of the Polish Academy of Sciences. Mathematics
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2021
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Vol. 69, no. 2
115--137
EN
We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope that some of them will turn out useful to achieve fast approximation from below of the hyperbolic Hausdorff dimension of Julia sets.
2
Content available remote On Lebesgue measure and Hausdorff dimension of Julia sets of real periodic points of renormalization
Dudko Artem
Bulletin of the Polish Academy of Sciences. Mathematics
|
2020
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Vol. 68, no. 2
151--168
EN
We give a new sufficient condition for the Julia set of a real analytic function which is a periodic point of renormalization to have Hausdorff dimension less than 2. This condition can be verified numerically. We present results of computer experiments suggesting that this condition is satisfied for real periodic points of renormalization with low periods. Our results support the conjecture that all real Feigenbaum maps have Julia sets of Hausdorff dimension less than 2.
3
Content available remote On the Exact Dimension of Mandelbrot Measure
Attia Najmeddine
Probability and Mathematical Statistics
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2019
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Vol. 39, Fasc. 2
299--314
EN
We develop, in the context of the boundary of a supercritical Galton-Watson tree, a uniform version of the argument used by Kahane (1987) on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets E(α) of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.
4
Content available remote On the Carrying Dimension of Occupation Measures for Self-Affine Random Fields
Kern Peter, Sönmez Ercan
Probability and Mathematical Statistics
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2019
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Vol. 39, Fasc. 2
459--479
EN
Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle (1984, 1988, 1990, 1991). Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.
5
Content available remote Dimension-theoretical results for a family of generalized continued fractions
Neunhäuserer J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2018
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Vol. 66, no. 2
115--122
EN
We find upper and lower estimates on the Hausdorff dimension of the set of real numbers which have coeffcients in a generalized continued fraction expansion that are bounded by a constant. As a consequence we prove a version of Jarník's theorem: the set of real numbers with bounded coeffcients in their generalized continued fraction representation has Hausdorff dimension one.
6
Content available remote Dimension results related to the St. Petersburg game
Kern P., Wedrich L.
Probability and Mathematical Statistics
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2014
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Vol. 34, Fasc. 1
97--117
EN
Let Sn be the total gain in n repeated St. Petersburg games. It is known that n−1(Sn − n log2 n) converges in distribution along certain geometrically increasing subsequences and its possible limiting random variables can be parametrized as Y (t) with t ∈ [1/2, 1]. We determine the Hausdorff and box-counting dimension of the range and the graph for almost all sample paths of the stochastic process {Y(t)}t∈[1/2, 1]. The results are compared to the fractal dimension of the corresponding limiting objects when gains are given by a deterministic sequence initiated by Hugo Steinhaus.
7
Content available Ubiquitiform in applied mechanics
Ou Z.-C., Li G.-Y., Duan Z.-P., Huang F-L.
Journal of Theoretical and Applied Mechanics
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2014
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Vol. 52 nr 1
37--46
EN
We demonstrate that a physical object in nature should not be described as a fractal, despite an ideal mathematical object, rather a ubiquitiform (a terminology coined here for a finite order self-similar or self-affine structure). It is shown mathematically that a ubiquitiform must be of integral dimension, and that the Hausdorff dimension of the initial element of a fractal changes abruptly at the point at infinity, which results in divergence of the integral dimensional measure of the fractal and makes the fractal approximation to a ubiquitiform unreasonable. Therefore, instead of the existing fractal theory in applied mechanics, a new type of ubiquitiformal one is needed.
8
Content available remote Non-Typical Points for β-Shifts
Färm D., Persson T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
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Vol. 61, no 2
123--132
EN
We study sets of non-typical points under the map fβ↦βx mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β>1.541 found in the above paper can be removed.
9
Content available remote On fractal dimension estimation
Pánek D., Kropik P., Predota A.
Przegląd Elektrotechniczny
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2011
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R. 87, nr 5
120-122
EN
The paper deals with an algorithm for Hausdorff dimension estimation based on box-counting dimension calculation. The main goal of the paper is to propose a new approach to box-counting dimension calculation with less computational demands.
PL
W artykule opisano algorytm estymacji wymiaru Hausdorffa oparty o wyznaczanie wymiaru pudełkowego. Głównym celem pracy jest zaproponowanie nowego podejścia do wyznaczania wymiaru pudełkowego, które ma znacznie mniejszą złożoność obliczeniową.
10
Content available remote Subadditive pressure for ifs with triangular maps
Barany B.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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Vol. 57, no 3-4
263-278
EN
We investigate properties of the zero of the subadditive pressure which is a most important tool to estimate the Hausdorff dimension of the attractor of a non-conformal iterated function system (IFS) . Our result is a generalization of the main results of Miao and Falconer [Fractals 15 (2007)] and Manning and Simon [Nonlinearity 20 (2007)].
11
Content available remote Infinite iterated function systems depending on a parameter
Jaksztas L.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
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Vol. 55, no 2
105-122
EN
This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets Jo,σ for the map fo(z) = x2 + 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 Jo,σ, given by Urbański and Zinsmeister. The closure of the limit set of our IFS {φ[...]} 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 Jo,σ. The parameter a determines the diameter of the largest circle, and therefore the diameters of other circles. We prove that for all parameters a except possibly for a set without accumulation points, for all appropriate t > 1 the sum of the tth powers of the diameters of the images of the largest circle under the maps of the IFS depends on the parameter σ. This is the first step to verifying the conjectured dependence of the pressure and Hausdorff dimension on a for our model and for Jo,σ.
12
Content available remote Generic properties of continuous iterated function systems with place dependent probabilities
Bielaczyc T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
|
Vol. 55, no 1
81-96
EN
It is shown that for a typical continuous learning system denned on a compact convex subset of R[sup]n the Hausdorff dimension of its invariant measure is equal to zero.
13
Content available remote On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers
Przytycki F.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
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Vol. 54, no 1
41-52
EN
We prove that the hyperbolic Hausdorff dimension of Fr Ω, the boundary of the simply connected immediate basin of attraction Ω to an attracting periodic point of a rational mapping of the Riemann sphere, which is not a finite Blaschke product in some holomorphic coordinates, or a 2 : 1 factor of a Blaschke product, is larger than 1. We prove a "local version" of this theorem, for a boundary repelling to the side of the domain. The results extend an analogous fact for polynomials proved by A. Zdunik and relies on the theory elaborated by M. Urbanski, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in δΩ constructed in [P2]. To reach our results, we deal with a quasi-repeller, i.e. the limit set for a geometric coding tree, and prove that the hyperbolic Hausdorff dimension of the limit set is larger than the Hausdorff dimension of the projection via the tree of any Gibbs measure for a Holder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Holder potentials on Julia sets.
14
Content available remote On the Hausdorff dimension of topological subspaces
Szarek T., Ślęczka M.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
Vol. 54, no 3-4
201-205
EN
It is shown that every Polish space X with dimT X ≥ d admits a compact subspace Y such that dimH Y ≥ d where dimT and dimH denote the topological and Hausdorff dimensions, respectively.
15
Content available remote Contracting-on-average Baker maps
Rams M.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
Vol. 54, no 3-4
219-229
EN
We estimate from above and below the Hausdorff dimension of SRB measure for contracting-on-average baker maps.
16
Content available remote On a dimension of measures
Lasota A., Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2002
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Vol. 50, no 2
221-235
EN
Using the notion of the Levy concentration function we discuss a definition of the dimension for probability measures. This dimension is strongly connected with the correlation dimension of measures and with the Hausdorff dimension of sets. Moreover, we calculate some bounds of this dimension for measures generated by Iterated Function Systems and by a partial differential equation.
17
Content available remote Hausdorff dimension theorems for self-similar Markov processes
Liu Luqin, Xiao Yimin
Probability and Mathematical Statistics
|
1998
|
Vol. 18, Fasc. 2
369--383
EN
Let X(t) (t ∈R+) be an α-self-similar Markov process on Rd or Rd+. The Hausdorff dimension of the image, graph and zero set of X (t) are obtained under certain mild conditions. Similar results are also proved for a class of elliptic diffusions.
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