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.
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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.
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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.
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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.
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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.
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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.
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.
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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.
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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.
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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)].
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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.
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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.
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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.
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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.
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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.
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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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