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1. CJM 2011 (vol 63 pp. 481)
The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal
representations of the ample or KÃ¤hler cone for surfaces in a
certain class of $K3$ surfaces. The class includes surfaces
described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a
sufficiently large number field $K$ that have a line parallel to
one of the axes and have Picard number four. We relate the
Hausdorff dimension of this fractal to the asymptotic growth of
orbits of curves under the action of the surface's group of
automorphisms. We experimentally estimate the Hausdorff dimension
of the fractal to be $1.296 \pm .010$.
Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 |
2. CJM 2011 (vol 63 pp. 648)
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps |
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps We set up a framework for computing the spectral dimension of a class of one-dimensional
self-similar measures that are defined by iterated function systems
with overlaps and satisfy a family of second-order self-similar
identities. As applications of our result we obtain the spectral dimension
of important measures such as the infinite Bernoulli convolution
associated with the golden ratio and convolutions of Cantor-type measures.
The main novelty of our result is that the iterated function systems
we consider are not post-critically finite and do not satisfy the
well-known open set condition.
Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75 |
3. CJM 2010 (vol 62 pp. 1182)
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces
is defined as a Haar wavelet decomposition, with the coefficients
depending on a fixed parameter $\beta>0$. On its support $I_0=[0,
1]^n$, $f(x)$ can be expressed by the binary expansions of the
coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb
R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for
$\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a
John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a self-affine function. It is continuous almost
everywhere and discontinuous at all dyadic points inside $I_0$. In
addition, it is not monotone along any coordinate direction in any
small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from
$I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the
graph of $f$ has a non-integer fractal dimension $n+1-\beta$.
Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension Categories:42B35, 42C10, 30D50, 28A80 |
4. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
5. CJM 1999 (vol 51 pp. 673)
Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of $\R^d$ formed in a manner
analogous to the construction of the Sierpinski carpet. We prove a
uniform Harnack inequality for positive harmonic functions; study
the heat equation, and obtain upper and lower bounds on the heat
kernel which are, up to constants, the best possible; construct a
locally isotropic diffusion $X$ and determine its basic properties;
and extend some classical Sobolev and Poincar\'e inequalities to
this setting.
Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions Categories:60J60, 60B05, 60J35 |