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 $E_{f}({0})$

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.Comment: 34 pages, 6 figure

Tensor Products of AC* Charges and AC Radon Measures Are Not Always AC* Charges

AbstractIn this note we give an example of an AC* charge, F, on R and an absolutely continuous Radon measure μ on R such that F⊗μ is not an AC* charge on R2

A Valós Analízis Dinamikai és Geometriai Mértékelméleti Vonatkozásai = Dynamical Systems, Geometric Measure Theoretic Aspects of Real Analysis

A pályázat futamideje alatt megjelentek a C. E. Weil gradiensproblémáját és I. Assani Ergodelméleti számolásproblémája megoldását tartalmazó cikkek, az utóbbi cikk társszerzői I. Assani és D. Mauldin. Egy ergodelméleti híres megoldatlan problémára választ adva sikerült olyan sorozatot konstruálnom, melyben a hézagok végtelenbe tartanak, de mégis teljesül rá a pontonkénti Ergodtétel. J. Bourgain egy még ennél is híresebb, négyzetek mentén vett ergodikus átlagokra vonatkozó problémájával kapcsolatban pedig hosszú évek munkájával sikerült meggyőznünk a nemzetközi tudományos közvéleményt konstrukciónk helyességéről. I. Assanival két Fürstenberg átlagokhoz kapcsolódó maximális operátorokra vonatkozó cikket készítettünk. Ezek is előrehaladást jelentenek a J. Bourgain eredményeivel kapcsolatos problémakörben. Két ELTÉs doktoranduszhallgatókkal közösen írt cikkben, pedig tipikus folytonos függvények mikrotangens halmazaival és egyértelműségű halmazaival kapcsolatban értünk el eredményeket. Egy munkámban pedig fraktálfüggvények szinthalmazaira, grafikonjaikon levő irreguláris halmazokra vonatkozó tételeket bizonyítottam. Elkészítettem és 2007-ben megvédtem a pályázat témakörével megegyező területet vizsgáló MTA doktori értekezésem. A pályázat részleges támogatásával szerveztem az "M60 A miniconference in Real Analysis" konferenciát. | During this project papers containing the solutions of the gradient problem of C. E. Weil and the counting problem of I. Assani got published, my coauthors on the latter paper were I. Assani and D. Mauldin. Answering a famous unsolved problem in Ergodic Theory I have managed to construct a sequence with gaps converging to infinity, but for which the pointwise Ergodic Theorem holds. With respect to an even more famous problem of J. Bourgain concerning ergodic averages along the squares we have managed to convince the scientific "general public" that our construction works. We prepared two papers with I. Assani about a maximal operator related to Furstenberg averages. These papers also contain progress related to results of J. Bourgain. In two joint papers written with Ph. D. students of our university we studied micro tangent sets and sets of univalence of typical continuous functions. In another paper I proved theorems concerning level sets and irregular sets on the graphs of fractal functions. I prepared and succesfully defended in 2007 my dissertation for the degree of "Doctor of Sciences of the Hungarian Academy of Sciences". The topic of this dissertation coincides with that of this research project. With partial support of this research grant I have organized the conference: "M60 A miniconference in Real Analysis"

Generic Birkhoff Spectra

Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ and ${\sigma}$ is the one-sided shift. The Birkhoff spectrum $\displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},$ where $\dim_{H}$ is the Hausdorff dimension. It is well-known that the support of $S_{f}( {\alpha})$ is a bounded and closed interval $L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]$ and $S_{f}( {\alpha})$ on $L_{f}$ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $f\in C( {\Omega})$ in the sense of Baire category. For a dense set in $C( {\Omega})$ the spectrum is not continuous on ${\mathbb {R}}$, though for the generic $f\in C( {\Omega})$ the spectrum is continuous on ${\mathbb {R}}$, but has infinite one-sided derivatives at the endpoints of $L_{f}$. We give an example of a function which has continuous $S_{f}$ on ${\mathbb {R}}$, but with finite one-sided derivatives at the endpoints of $L_{f}$. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $f$ and $g$ are close in $C( {\Omega})$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from neighborhoods of the endpoints.Comment: Revised version after the referee's repor

On series of translates of positive functions II

AbstractIn this paper we continue our investigation of series of the form ∑λ ∈ Λ ƒ(x + λ). Given a sequence of natural numbers n1 < n2 < … we are interested in sets Λ of the form where 0 < α < 1. In case α = 1q, where q > 1 is an integer, there is a zero-one law showing that for every measurable the above sum either converges almost everywhere or diverges almost everywhere. However, for any other value of α ∈ (0, 1) there is no such zero-one law

Box dimension of generic H\"older level sets

Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections" "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. Finer information might be obtained by considering the Hausdorff dimension of level sets of generic $1$-H\"older-$\alpha$ functions, which has a stronger dependence on the geometry of the fractal, as displayed in our previous papers. In this paper, we extend our investigations to the lower and upper box-counting dimension as well: while the former yields results highly resembling the ones about Hausdorff dimension of level sets, the latter exhibits a different behaviour. Instead of "finding narrow-cross sections", results related to upper box-counting dimension try to "measure" how much level sets can spread out on the fractal, how widely the generic function can "oscillate" on it. Key differences are illustrated by giving estimates concerning the Sierpi\'nski triangle

Big and little Lipschitz one sets

Given a continuous function $f: {{\mathbb R}}\to {{\mathbb R}}$ we denote the so-called "big Lip" and "little lip" functions by ${{\mathrm {Lip}}} f$ and ${{\mathrm {lip}}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$ or ${{\mathrm {Lip}}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_\delta$ and UDT then there exists a continuous function $f$ satisfying ${{\mathrm {Lip}}} f =\mathbf{1}_E$, that is, $E$ is a ${{\mathrm {Lip}}} 1$ set. In the other direction we show that every ${{\mathrm {Lip}}} 1$ set is $G_\delta$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{{\delta}}$ sets which are not ${{\mathrm {Lip}}} 1$. We say that a set $E\subset \mathbb{R}$ is ${{\mathrm {lip}}} 1$ if there is a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${{\mathrm {lip}}} 1$ set is a strongly one-sided dense $F_\sigma$ set.Comment: This is the final preprint version accepted to appear in European Journal of Mathematic