Expand all Collapse all | Results 1 - 22 of 22 |
1. CMB 2014 (vol 58 pp. 71)
Limited Sets and Bibasic Sequences Bibasic sequences are used to study relative weak compactness
and relative norm compactness of limited sets.
Keywords:limited sets, $L$-sets, bibasic sequences, the Dunford-Pettis property Categories:46B20, 46B28, 28B05 |
2. CMB 2012 (vol 57 pp. 240)
Addendum to ``Limit Sets of Typical Homeomorphisms'' Given an integer $n \geq 3$,
a metrizable compact topological $n$-manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$-almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
3. CMB 2011 (vol 56 pp. 326)
Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$ We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms
of some fractal measures.
Keywords:Fourier transforms of fractal measures, Fourier restriction Categories:42B10, 28A12 |
4. CMB 2011 (vol 56 pp. 354)
The Sizes of Rearrangements of Cantor Sets A linear Cantor set $C$ with zero Lebesgue measure is associated with
the countable collection of the bounded complementary open intervals. A
rearrangment of $C$ has the same lengths of its complementary
intervals, but with different locations. We study the Hausdorff and packing
$h$-measures and dimensional properties of the set of all rearrangments of
some given $C$ for general dimension functions $h$. For each set of
complementary lengths, we construct a Cantor set rearrangement which has the
maximal Hausdorff and the minimal packing $h$-premeasure, up to a constant.
We also show that if the packing measure of this Cantor set is positive,
then there is a rearrangement which has infinite packing measure.
Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set Categories:28A78, 28A80 |
5. CMB 2011 (vol 56 pp. 292)
Quasisymmetrically Minimal Moran Sets M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor
sets of Hausdorff dimension $1$, where at the $k$-th set one removes
from each interval $I$ a certain number $n_{k}$ of open subintervals
of length $c_{k}|I|$, leaving $(n_{k}+1)$ closed subintervals of
equal length. Quasisymmetrically Moran sets of Hausdorff dimension $1$
considered in the paper are more general than uniform Cantor sets in
that neither the open subintervals nor the closed subintervals are
required to be of equal length.
Keywords:quasisymmetric, Moran set, Hausdorff dimension Categories:28A80, 54C30 |
6. CMB 2011 (vol 55 pp. 830)
Almost Everywhere Convergence of Convolution Measures Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $(X,\mathcal{B},m)$ a probability
space and $\tau$ an invertible, measure preserving transformation.
This paper deals with the almost everywhere convergence in $\textrm{L}^1(X)$ of a
sequence of operators of weighted averages. Almost everywhere convergence follows
once we obtain an appropriate maximal estimate and once we provide
a dense class where convergence holds almost everywhere.
The weights are given by convolution products of members of a sequence of probability
measures $\{\nu_i\}$ defined on $\mathbb{Z}$.
We then exhibit cases of such averages where convergence fails.
Category:28D |
7. CMB 2011 (vol 55 pp. 723)
First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
below.
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.
Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow Categories:53C23, 28A35, 49Q20, 58A35 |
8. CMB 2011 (vol 55 pp. 225)
Limit Sets of Typical Homeomorphisms Given an integer $n \geq 3$, a metrizable compact
topological $n$-manifold $X$ with boundary, and a finite positive Borel
measure $\mu$ on $X$, we prove that for the typical homeomorphism
$f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
non-sensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
9. CMB 2011 (vol 55 pp. 815)
Restricted Radon Transforms and Projections of Planar Sets We establish a mixed norm estimate for the Radon transform in
$\mathbb{R}^2$ when the set of directions has fractional dimension.
This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to
a conjecture analogous to a well-known conjecture of Furstenberg.
Categories:44A12, 28A78 |
10. CMB 2011 (vol 54 pp. 706)
Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve We provide a simple example showing that the tangential derivative of a
continuous function $\phi$
can vanish everywhere along a curve while the variation of $\phi$ along
this curve is nonzero. We give additional regularity conditions on the curve
and/or the function that prevent this from happening.
Categories:26A24, 28A15 |
11. CMB 2010 (vol 54 pp. 172)
Measures with Fourier Transforms in $L^2$ of a Half-space
We prove that if the Fourier transform of a compactly supported
measure is in $L^2$ of a half-space, then the measure is
absolutely continuous to Lebesgue measure. We then show how this
result can be used to translate information about the
dimensionality of a measure and the decay of its Fourier
transform into geometric information about its support.
Categories:42B10, 28A75 |
12. CMB 2009 (vol 52 pp. 105)
Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket We prove there exist exponentially decaying generalized eigenfunctions
on a blow-up of the Sierpinski gasket with boundary. These are used
to show a Borel-type theorem, specifically that for a prescribed jet
at the boundary point there is a smooth function having that jet.
Categories:28A80, 31C45 |
13. CMB 2006 (vol 49 pp. 247)
A Szpilrajn--Marczewski Type Theorem for Concentration Dimension on Polish Spaces Let $X$ be a Polish space.
We will prove that
$$
\dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to
} X\},
$$
where $\dim_L X$ and $\dim_T X$ stand
for the concentration dimension and
the topological dimension of $X$, respectively.
Keywords:Hausdorff dimension, topological dimension, LÃ©vy concentration function, concentration dimension Categories:11K55, 28A78 |
14. CMB 2006 (vol 49 pp. 203)
The Ergodic Hilbert Transform for Admissible Processes It is shown that the ergodic Hilbert transform
exists for a class of bounded symmetric admissible processes
relative to invertible measure preserving transformations. This
generalizes the well-known result on the existence of the ergodic
Hilbert transform.
Keywords:Hilbert transform, admissible processes Categories:28D05, 37A99 |
15. CMB 2004 (vol 47 pp. 168)
Kolakoski-$(3,1)$ Is a (Deformed) Model Set Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding bi-infinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski-$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.
Categories:52C23, 37B10, 28A80, 43A25 |
16. CMB 2002 (vol 45 pp. 123)
Uniform Distribution in Model Sets We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 |
17. CMB 2001 (vol 44 pp. 429)
Ergodic Rotations of Nilmanifolds Conjugate to Their Inverses In answer to a question posed in \cite{G}, we give sufficient
conditions on a Lie nilmanifold so that any ergodic rotation of the
nilmanifold is metrically conjugate to its inverse. The condition is
that the Lie algebra be what we call quasi-graded, and is weaker than
the property of being graded. Furthermore, the conjugating map can be
chosen to be an involution. It is shown that for a special class of
groups, the condition of quasi-graded is also necessary. In certain
examples there is a continuum of conjugacies.
Categories:28Dxx, 22E25 |
18. CMB 2001 (vol 44 pp. 61)
The Inequalities for Polynomials and Integration over Fractal Arcs The paper is dealing with determination of the integral $\int_{\gamma}
f \,dz$ along the fractal arc $\gamma$ on the complex plane by terms
of polynomial approximations of the function~$f$. We obtain
inequalities for polynomials and conditions of integrability for
functions from the H\"older, Besov and Slobodetskii spaces.
Categories:26B15, 28A80 |
19. CMB 2000 (vol 43 pp. 157)
A Larger Class of Ornstein Transformations with Mixing Property We prove that Ornstein transformations are almost surely totally
ergodic provided only that the cutting parameter is not bounded.
We thus obtain a larger class of Ornstein transformations with the
mixing property.
Categories:28D05, 47A35 |
20. CMB 1999 (vol 42 pp. 291)
Spaces of Quasi-Measures We give a direct proof that the space of Baire quasi-measures on a
completely regular space (or the space of Borel quasi-measures on a
normal space) is compact Hausdorff. We show that it is possible for
the space of Borel quasi-measures on a non-normal space to be
non-compact. This result also provides an example of a Baire
quasi-measure that has no extension to a Borel quasi-measure. Finally,
we give a concise proof of the Wheeler-Shakmatov theorem, which states
that if $X$ is normal and $\dim(X) \le 1$, then every
quasi-measure on $X$ extends to a measure.
Category:28C15 |
21. CMB 1998 (vol 41 pp. 41)
On the Clarke subdifferential of an integral functional on $L_p$, $1\leq p < \infty$ Given an integral functional defined on $L_p$, $1 \leq p <\infty$,
under a growth condition we give an upper bound of the Clarke
directional derivative and we obtain a nice inclusion between the
Clarke subdifferential of the integral functional and the set of
selections of the subdifferential of the integrand.
Keywords:Integral functional, integrand, epi-derivative Categories:28A25, 49J52, 46E30 |
22. CMB 1997 (vol 40 pp. 3)
La formule de Cauchy sur la longueur d'une courbe Pour toute courbe rectifiable du plan, nous d\'emontrons la formule
de Cauchy relative \`a sa longueur. La formule est donn\'ee sous deux
formes: comme int\'egrale de la variation totale des projections de la
courbe dans les diverses directions et comme int\'egrale double du
nombre de rencontres de la courbe avec une droite quelconque du plan.
We give a general proof of the Cauchy formula about the length of a
plane curve. The formula is given in two ways: as the integral of the
variation of orthogonal projections of the curve, and as a double
integral of the number of intersections of the curve with an arbitrary
line of the plane.
Keywords:longueur, variation bornÃ©e, gÃ©omÃ©trie intÃ©grale Categories:28A75, 28A45 |