137 research outputs found
A Nonlinear Mercerian Theorem
AbstractWe show that convergence of x(t) as tââ may be deduced from the limiting behavior of certain functions involving x(t) and its CesĂ ro averages. Dynamical systems methods are used to derive this âMercerian-typeâ result
Small union with large set of centers
Let be a fixed set. By a scaled copy of around
we mean a set of the form for some .
In this survey paper we study results about the following type of problems:
How small can a set be if it contains a scaled copy of around every point
of a set of given size? We will consider the cases when is circle or sphere
centered at the origin, Cantor set in , the boundary of a square
centered at the origin, or more generally the -skeleton () of an
-dimensional cube centered at the origin or the -skeleton of a more
general polytope of .
We also study the case when we allow not only scaled copies but also scaled
and rotated copies and also the case when we allow only rotated copies
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
Multifractal properties of convex hulls of typical continuous functions
We study the singularity (multifractal) spectrum of the convex hull of the
typical/generic continuous functions defined on . We denote by
the set of points at which has a pointwise H\"older exponent equal to . Let
be the convex hull of the graph of , the concave function on the top
of is denoted by and denotes the convex function on
the bottom of .
We show that there is a dense subset such that for the following properties are
satisfied. For the functions and
coincide only on a set of zero Hausdorff dimension, the functions are continuously differentiable on , equals the boundary of ,
, and if
Self-similar sets: projections, sections and percolation
We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.Postprin
The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence
One of the most widely used methods for eigenvalue computation is the
iteration with Wilkinson's shift: here the shift is the eigenvalue of the
bottom principal minor closest to the corner entry. It has been a
long-standing conjecture that the rate of convergence of the algorithm is
cubic. In contrast, we show that there exist matrices for which the rate of
convergence is strictly quadratic. More precisely, let be the matrix having only two nonzero entries and let
be the set of real, symmetric tridiagonal matrices with the same spectrum
as . There exists a neighborhood of which is
invariant under Wilkinson's shift strategy with the following properties. For
, the sequence of iterates exhibits either strictly
quadratic or strictly cubic convergence to zero of the entry . In
fact, quadratic convergence occurs exactly when . Let be
the union of such quadratically convergent sequences : the set has
Hausdorff dimension 1 and is a union of disjoint arcs meeting at
, where ranges over a Cantor set.Comment: 20 pages, 8 figures. Some passages rewritten for clarit
A process very similar to multifractional Brownian motion
In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is
obtained by replacing the constant parameter of the fractional Brownian
motion (fBm) by a smooth enough functional parameter depending on the
time . Here, we consider the process obtained by replacing in the
wavelet expansion of the fBm the index by a function depending on
the dyadic point . This process was introduced in Benassi et al (2000)
to model fBm with piece-wise constant Hurst index and continuous paths. In this
work, we investigate the case where the functional parameter satisfies an
uniform H\"older condition of order \beta>\sup_{t\in \rit} H(t) and ones
shows that, in this case, the process is very similar to the mBm in the
following senses: i) the difference between and a mBm satisfies an uniform
H\"older condition of order ; ii) as a by product, one
deduces that at each point the pointwise H\"older exponent of is
and that is tangent to a fBm with Hurst parameter .Comment: 18 page
Slicing Sets and Measures, and the Dimension of Exceptional Parameters
We consider the problem of slicing a compact metric space \Omega with sets of
the form \pi_{\lambda}^{-1}\{t\}, where the mappings \pi_{\lambda} \colon
\Omega \to \R, \lambda \in \R, are \emph{generalized projections}, introduced
by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that
\Omega has Hausdorff dimension strictly greater than one, what is the dimension
of the 'typical' slice \pi_{\lambda}^{-1}{t}, as the parameters \lambda and t
vary. In the special case of the mappings \pi_{\lambda} being orthogonal
projections restricted to a compact set \Omega \subset \R^{2}, the problem
dates back to a 1954 paper by Marstrand: he proved that for almost every
\lambda there exist positively many such that \dim
\pi_{\lambda}^{-1}{t} = \dim \Omega - 1. For generalized projections, the same
result was obtained 50 years later by J\"arvenp\"a\"a, J\"arvenp\"a\"a and
Niemel\"a. In this paper, we improve the previously existing estimates by
replacing the phrase 'almost all \lambda' with a sharp bound for the dimension
of the exceptional parameters.Comment: 31 pages, three figures; several typos corrected and large parts of
the third section rewritten in v3; to appear in J. Geom. Ana
Lines Missing Every Random Point
We prove that there is, in every direction in Euclidean space, a line that
misses every computably random point. We also prove that there exist, in every
direction in Euclidean space, arbitrarily long line segments missing every
double exponential time random point.Comment: Added a section: "Betting in Doubly Exponential Time.
Hausdorff dimension of operator semistable L\'evy processes
Let be an operator semistable L\'evy process in \rd
with exponent , where is an invertible linear operator on \rd and
is semi-selfsimilar with respect to . By refining arguments given in
Meerschaert and Xiao \cite{MX} for the special case of an operator stable
(selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we
determine the Hausdorff dimension of the partial range in terms of the
real parts of the eigenvalues of and the Hausdorff dimension of .Comment: 23 page
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