In this paper, we give several pictorial fractal representations of the ample or Kähler 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 $1.296 \pm .010$.

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.

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.

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.