67 research outputs found
Multifractal properties of typical convex functions
We study the singularity (multifractal) spectrum of continuous convex
functions defined on . Let be the set of points at which
has a pointwise exponent equal to . We first obtain general upper bounds
for the Hausdorff dimension of these sets , for all convex functions
and all . We prove that for typical/generic (in the sense of
Baire) continuous convex functions , one has for all and in addition, we obtain that the set is empty if . Also, when is typical,
the boundary of belongs to
Measures and functions with prescribed homogeneous multifractal spectrum
In this paper we construct measures supported in with prescribed
multifractal spectrum. Moreover, these measures are homogeneously multifractal
(HM, for short), in the sense that their restriction on any subinterval of
has the same multifractal spectrum as the whole measure. The spectra
that we are able to prescribe are suprema of a countable set of step
functions supported by subintervals of and satisfy for all
. We also find a surprising constraint on the multifractal spectrum
of a HM measure: the support of its spectrum within 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 . 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 and is the
one-sided shift. The Birkhoff spectrum where is the
Hausdorff dimension. It is well-known that the support of is
a bounded and closed interval
and on is concave and upper semicontinuous. We are
interested in possible shapes/properties of the spectrum, especially for
generic/typical in the sense of Baire category. For a dense
set in the spectrum is not continuous on ,
though for the generic the spectrum is continuous on , but has infinite one-sided derivatives at the endpoints of
. We give an example of a function which has continuous on , but with finite one-sided derivatives at the endpoints of
. The spectrum of this function can be as close as possible to a
"minimal spectrum". We use that if two functions and are close in then and are close on 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, . 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
-H\"older- 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 we denote the
so-called "big Lip" and "little lip" functions by and respectively}. In this paper we are interested in the
following question. Given a set is it possible to
find a continuous function such that or
?
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 is
and UDT then there exists a continuous function satisfying , that is, is a
set.
In the other direction we show that every set is
and weakly dense. We also show that the converse of this statement
is not true, namely that there exist weakly dense sets which are
not .
We say that a set is if there is
a continuous function such that . We
introduce the concept of strongly one-sided density and show that every
set is a strongly one-sided dense set.Comment: This is the final preprint version accepted to appear in European
Journal of Mathematic
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