arXiv subjects

Zoltán Buczolich

Publications and source records attributed to Zoltán Buczolich.

27 records · Page 2Linked to original sources

arXiv2019

Isentropes and Lyapunov exponents

We consider skew tent maps $T_{α, β}(x)$ such that $( α, β)\in[0,1]^{2}$ is the turning point of $T {_ { α, β}}$, that is, $T_{α, β}=\frac{β}{α}x$ for $0\leq x \leq α$ and $T_{α, β}(x)=\frac{β}{1- α}(1-x)$ for $ α<x\leq 1$. We denote by $ {\underline {M}}=K( α, β)$ the kneading sequence of $T {_ { α, β}}$, by $h( α, β)$ its topological entropy and $Λ=Λ_{α,β}$ denotes its Lyapunov exponent. For a given kneading squence $ {\underline {M}}$ we consider isentropes (or equi-topological entropy, or equi-kneading curves), $( α,Ψ_{\underline {M}}( α))$ such that $K( α,Ψ_{\underline {M}}( α))= {\underline {M}}$. On these curves the topological entropy $h( α,Ψ_{\underline {M}}( α))$ is constant. We show that $Ψ_{\underline {M}}'( α)$ exists and the Lyapunov exponent $Λ_{α,β}$ can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function $ { Θ}_{\underline {M}}$, a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.

math.DS↗

arXiv2018

Type $1$ and $2$ sets for series of translates of functions

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $1$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= {\mathbb R}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= {\mathbb R}$ modulo sets of measure zero. If $ Λ$ is not type $1$ we say that $ Λ$ is type 2. The exact characterization of type $1$ and type $2$ sets is not known. In this paper we continue our study of the properties of type $1$ and $2$ sets. We discuss sub and supersets of type $1$ and $2$ sets and we give a complete and simple characterization of a subclass of dyadic type $1$ sets. We discuss the existence of type $1$ sets containing infinitely many algebraically independent elements. Finally, we consider unions and Minkowski sums of type $1$ and $2$ sets.

math.CA↗

arXiv2018

On sets where $\operatorname{lip} f$ is finite

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r} \frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{σδ}$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost everywhere.

math.CA↗

arXiv2018

Random constructions for translates of non-negative functions

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $2$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ does not satisfy a zero-one law. This means that we can find a non-negative measurable "witness function" $f: {\mathbb R}\to [0,+ {\infty})$ such that both the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}$ and its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}$ are of positive Lebesgue measure. If $ Λ$ is not type $2$ we say that $ Λ$ is type $1$. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set $ Λ$ if we randomly delete its elements. Motivated by results concerning weighted sums $\sum c_n f(nx)$ and the Khinchin conjecture, we also discuss some results about weighted sums $\sum_{n=1}^{\infty}c_n f(x+λ_n)$.

math.CA↗

arXiv2018

On series of translates of positive functions III

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is of type 1 if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: { {\mathbb R}}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= { {\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= { {\mathbb R}}$ modulo sets of measure zero. If $ Λ$ is not of type 1 we say that $ Λ$ is of type 2. In this paper we show that there is a universal $ Λ$ with gaps monotone decreasingly converging to zero such that for any open subset $G \subset { {\mathbb R}}$ one can find a characteristic function $f_{G}$ such that $G \subset D(f_G, Λ)$ and $C(f_G, Λ)= { {\mathbb R}} {\setminus} G$ modulo sets of measure zero. We also consider the question whether $C(f, Λ)$ can contain non-degenerate intervals for continuous functions when $D(f, Λ)$ is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.

math.CA↗

arXiv2017

Multifractal properties of typical convex functions

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the Hausdorff dimension of these sets $E_f(h)$, for all convex functions $f$ and all $h\geq 0$. We prove that for typical/generic (in the sense of Baire) continuous convex functions $f:[0,1]^{d}\to \mathbb{R} $, one has $\dim E_f(h) =d-2+h$ for all $h\in[1,2],$ and in addition, we obtain that the set $ E_f({h} )$ is empty if $h\in (0,1)\cup (1,+\infty)$. Also, when $f$ is typical, the boundary of $[0,1]^{d}$ belongs to $E_{f}({0})$.

math.CA↗

arXiv2015

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension. We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc. As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually `thick' families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension. Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.

math.CA↗

arXiv2013

Measures and functions with prescribed homogeneous multifractal spectrum

In this paper we construct measures supported in $[0,1]$ with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of $[0,1]$ has the same multifractal spectrum as the whole measure. The spectra $f$ that we are able to prescribe are suprema of a countable set of step functions supported by subintervals of $[0,1]$ and satisfy $f(h)\leq h$ for all $h\in [0,1]$. We also find a surprising constraint on the multifractal spectrum of a HM measure: the support of its spectrum within $[0,1]$ must be an interval. This result is a sort of Darboux theorem for multifractal spectra of measures. This result is optimal, since we construct a HM measure with spectrum supported by $[0,1] \cup {2}$. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.

math.CA↗

arXiv2010

Typical Borel measures on $[0,1]d$ satisfy a multifractal formalism

In this article, we prove that in the Baire category sense, measures supported by the unit cube of $\R^d$ typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures $μ$. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension $d$ at exponent $d$, and it indeed coincides with the Legendre transform of the $L^q$-spectrum associated with typical measures $μ$.

math-ph↗