Hausdorff Dimension Of Level Sets Research Articles

Generic Hölder level sets and fractal conductivity

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the “thickness/narrow cross-sections” of a “network” corresponding to a fractal set, F. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic 1-Hölder-α functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations give better insight and illustrate why can one think of these generic 1-Hölder-α level sets as something measuring “thickness/narrow cross-sections/conductivity” of a fractal “network”.We also give an example for the phenomenon which we call phase transition for D∗(α,F). This roughly means that for a certain lower range of αs only the geometry of F determines D∗(α,F) while for larger values the Hölder exponent, α also matters.

Generic Hölder level sets on fractals

Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic 1-Hölder-α functions defined on fractals. This is related to some sort of “thickness”, “conductivity” properties of fractals. The main concept of our paper is D⁎(α,F) which is the essential supremum of the Hausdorff dimensions of the level sets of a generic 1-Hölder-α function defined on the fractal F. We prove some basic properties of D⁎(α,F), we calculate its value for an example of a “thick fractal sponge”, we show that for connected self similar sets D⁎(α,F) it equals the Hausdorff dimension of almost every level in the range of a generic 1-Hölder-α function.

Hausdorff dimensions of level sets related to moving digit means

In this paper, we will introduce and study the lower moving digit mean B_(x) and the upper moving digit mean B¯(x) of x∈[0,1] in p-adic expansion, where p≥2 is an integer. Moreover, we will determine the Hausdorff dimensions of level setsB(α,β)={x∈I:B_(x)=α,B¯(x)=β},0≤α≤β≤p−1. This work is closely related to an early classic work of Besicovitch in the dimensional theory, and it generalizes the work from the ordinary digit means to the moving digit means.

Hausdorff dimension of level sets in Engel continued fraction

Hausdorff dimension of level sets in Engel continued fraction

The Hausdorff dimension of level sets described by Erdös–Rényi average

Let 0≤α≤1 and ϕ be an integer function defined on N∖{0} satisfying 1≤ϕ(n)≤n. Define the level setERϕ(α)={x∈[0,1]:limn→∞⁡An,ϕ(n)(x)=α}, where An,ϕ(n)(x) is the (n,ϕ(n))-Erdös–Rényi average of x∈[0,1]. In this paper, we will give descriptions for the Hausdorff dimension of ERϕ(α) under the assumption ϕ(n)→∞ as n→∞, which complement simultaneously an early classic result of Besicovitch and the new strong law of large number established by P. Erdös and A. Rényi. Moreover, for the case ϕ(n)=M ultimately, where M≥1 is an integer, the Hausdorff dimension of ERϕ(α) is also determined by us in the last section.

Multifractal analysis on the level sets described by moving averages

In this paper, the Hausdorff dimensions of level sets described by two kinds of moving averages are determined. The dissimilar results complement the work of Pfaffelhuber [Moving shift averages for ergodic transformation, Metrika22 (1975) 97–101] and del Junco and Steele [Moving averages of ergodic processes, Metrika24 (1977) 35–43], and reveal simultaneously that the two moving averages are of different convergence processes in the ergodic theory.

Hausdorff dimension of level sets of generalized Takagi functions

AbstractThis paper examines the Hausdorff dimension of the level setsf−1(y) of continuous functions of the form\begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*}where φ(x) is the distance fromxto the nearest integer, and for eachn, ωnis a {−1,1}-valued function which is constant on each interval [j/2n,(j+1)/2n),j=0,1,. . .,2n− 1. This class of functions includes Takagi's continuous but nowhere differentiable function. It is shown that the largest possible Hausdorff dimension off−1(y) is$\log ((9+\sqrt{105})/2)/\log 16\approx .8166$, but in case each ωnis constant, the largest possible dimension is 1/2. These results are extended to the intersection of the graph offwith lines of arbitrary integer slope. Furthermore, two natural models of choosing the signs ωn(x) at random are considered, and almost-sure results are obtained for the Hausdorff dimension of the zero set and the set of maximum points off. The paper ends with a list of open problems.

Multifractal formalism for Benedicks–Carleson quadratic maps

AbstractFor a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to ‘see’ sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

Hausdorff Dimension of the Set of Generic Points for Gibbs Measures

For a Gibbs measure on the configuration space of a finite spin lattice system, we find (in terms of entropy) the Hausdorff dimension of the set of generic points. Using this result, we evaluate the Hausdorff dimension of level sets for Birkhoff ergodic averages of some continuous functions on the configuration space.

Hausdorff dimensions of level sets of Rademacher series

Let { a n } n≥1 be the sequence of real numbers such that lim n→∞ a n=0 and ∑ n=1 ∞ |a n|=∞ . Consider the Rademacher series S(x)= ∑ n=1 ∞ a nR n(x) and its level sets E a ={ x∈[0,1]: S( x)= a} (for all a∈ R ). We prove that dim H E a=1 for any a∈ R , where dim H is Hausdorff dimension.

Hausdorff Measure and Level Sets of Typical Continuous Mappings in Euclidean Spaces

We determine the Hausdorff dimension of level sets and of sets of points of multiplicity for mappings in a residual subset of the space of all continuous mappings from ${\mathbb {R}^n}$ to ${\mathbb {R}^m}$.

Hausdorff measure and level sets of typical continuous mappings in Euclidean spaces

We determine the Hausdorff dimension of level sets and of sets of points of multiplicity for mappings in a residual subset of the space of all continuous mappings from R n {\mathbb {R}^n} to R m {\mathbb {R}^m} .

Hausdorff dimension of level sets of some Rademacher series

Hausdorff dimension of level sets of some Rademacher series