Expand all Collapse all | Results 1 - 4 of 4 |
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 2008 (vol 60 pp. 658)
Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps We study the case of an Axiom A holomorphic non-degenerate
(hence non-invertible) map $f\from\mathbb P^2
\mathbb C \to \mathbb P^2 \mathbb C$, where $\mathbb P^2 \mathbb C$
stands for the complex
projective space of dimension 2. Let $\Lambda$ denote a basic set for
$f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda$; we
denote by $\delta^s(x)$ the Hausdorff dimension of $W^s_r(x) \cap
\Lambda$, where $r$ is some fixed positive number and $W^s_r(x)$ is
the local stable manifold at $x$ of size $r$; $\delta^s(x)$ is called
\emph{the stable dimension at} $x$. Mihailescu and
Urba\'nski introduced a notion of inverse topological pressure,
denoted by $P^-$, which takes into consideration preimages of points.
Manning and McCluskey study the case of hyperbolic diffeomorphisms on
real surfaces and give formulas for Hausdorff dimension. Our
non-invertible situation is different here since the local unstable
manifolds are not uniquely determined by their base point, instead
they depend in general on whole prehistories of the base points. Hence
our methods are different and are based on using a sequence of inverse
pressures for the iterates of $f$, in order to give upper and lower
estimates of the stable dimension. We obtain an estimate of the
oscillation of the stable dimension on $\Lambda$. When each point $x$
from $\Lambda$ has the same number $d'$ of preimages in $\Lambda$,
then we show that $\delta^s(x)$ is independent
of $x$; in fact $\delta^s(x)$ is shown to be equal in this case with
the unique zero of the map $t \to P(t\phi^s - \log d')$. We also
prove the Lipschitz continuity of the stable vector spaces over
$\Lambda$; this proof is again different than the one for
diffeomorphisms (however, the unstable distribution is not always
Lipschitz for conformal non-invertible maps). In the end we include
the corresponding results for a real conformal setting.
Keywords:Hausdorff dimension, stable manifolds, basic sets, inverse topological pressure Categories:37D20, 37A35, 37F35 |
3. CJM 2002 (vol 54 pp. 1280)
Besov Spaces and Hausdorff Dimension For Some Carnot-CarathÃ©odory Metric Spaces We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$
satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a
unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian
$\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic
decomposition of the spaces is given. In consequence we get the distributional
characterization of the Hausdorff dimension of Borel subsets with the Haar measure
zero.
Keywords:Besov spaces, sub-elliptic operators, Carnot-CarathÃ©odory metric, Hausdorff dimension Categories:46E35, 43A15, 28A78 |
4. 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 |