The subject of fractal geometry has exploded over the past 40 years with the availability of computer generated images. It was seen early on that there are many interesting questions at the intersection of probability and fractal geometry. In this dissertation we will introduce two random models for constructing fractals and prove various facts about them.

Dans cette thèse, on étudie les sous-groupes boréliens des groupes de Lie et leur dimension de Hausdorff. Si G est un groupe de Lie nilpotent connexe, on construit dans G des sous-groupes de dimension de Hausdorff arbitraire, tandis que si G est semisimple compact, on démontre que la dimension de Hausdorff d'un sous-groupe borélien strict de G ne peut pas être arbitrairement proche de celle de G. / Given a Lie group G, we investigate the possible Hausdorff dimensions for a measurable subgroup of G. If G is a connected nilpotent Lie group, we construct measurable subgroups of G having arbitrary Hausdorff dimension, whereas if G is compact semisimple, we show that a proper measurable subgroup of G cannot have Hausdorff dimension arbitrarily close to the dimension of G.

Given a factor code [pi] from a shift of finite type X onto an irreducible sofic shift Y, and a fully supported ergodic measure v on Y we give an explicit upper bound on the number of ergodic measures on X which project to v and have maximal entropy among all measures in the fiber [pi]-1{v}. This bound is invariant under conjugacy. We relate this to an important construction for finite-to-one symbolic factor maps.

Soit G le groupe SO°(1, n) (n ≥ 3) ou PU(1, n) (n ≥ 2) et fixons une décomposition d'Iwasawa G = KAN. Soit ɼ un sous-groupe discret de G, que nous supposons Zariski-dense et de mesure de Bowen-Margulis-Sullivan finie. Lorsque G = SO°(1, n), nous étudions la géométrie de la mesure de Bowen-Margulis-Sullivan le long des sous-groupes fermés connexes de N, en lien avec la dichotomie de Mohammadi-Oh. Nous établissons des résultats déterministes sur la dimension des projections de la mesure de Patterson- Sullivan. Lorsque G = PU(1, n), nous relions la géométrie de la mesure de Bowen- Margulis-Sullivan le long du centre du groupe de Heisenberg au problème du calcul de la dimension de Hausdorff de l'ensemble limite relativement à la distance sphérique au bord. Nous calculons cette dimension pour certains groupes de Schottky. / Let G be the group SO° (1,n) (n ≥ 3) or PU(1, n) (n ≥ 2) and fix some Iwasawa decomposition G = KAN. Let ɼ be a discrete subgroup of G.We assume that ɼ is Zariski-dense with finite Bowen-Margulis-Sullivan measure. When G = SO°(1,n), we investigate the geometry of the Bowen-Margulis-Sullivan measure elong connected closed subgroups of N. This is related to the Mohammadi-Oh dichotomy. We then prove deterministic results on the dimension of projections of Patterson-Sullivan measure. When G = PU(1,n), we relate the geometry of Bowen-Margulis-Sullivan measure along the center of Heisenberg group to the problem of computing the Hausdorff dimension of the limit set with respect to the spherical metric on the boudary. We construct some Schottky subgroups for wich we are able to compute this dimension.

In this thesis, we investigate examples of random walks on free products of cyclic groups. Free products are groups that contain words constructed by concatenation with possible simplifications[20]. Mairesse in [17] proved that the harmonic measure on the boundary of these random walks has a Markovian Multiplicative structure (this is a class of Markov measures which requires fewer parameters than the usual Markov measures for its description ), and also showed how in the case of the harmonic measure these parameters can be found from Traffic Equations. Then Mairesse and Math ́eus in [20] continued investigation of these random walks and the associated Traffic Equations. They introduced the Stationary Traffic Equations for the situation when the measure is shift-invariant in addition to being μ-invariant. In this thesis, we review these developments as well as explicitly describe several concrete examples of random walks on free products, some of which are new.

In this dissertation we study graph directed Markov systems (GDMS) and limit sets associated with these systems. Given a GDMS S, by the Hausdorff dimension spectrum of S we mean the set of all positive real numbers which are the Hausdorff dimension of the limit set generated by a subsystem of S. We say that S has full Hausdorff dimension spectrum (full HD spectrum), if the dimension spectrum is the interval [0, h], where h is the Hausdorff dimension of the limit set of S. We give necessary conditions for a finitely primitive conformal GDMS to have full HD spectrum. A GDMS is said to be regular if the Hausdorff dimension of its limit set is also the zero of the topological pressure function. We show that every number in the Hausdorff dimension spectrum is the Hausdorff dimension of a regular subsystem. In the particular case of a conformal iterated function system we show that the Hausdorff dimension spectrum is compact. We introduce several new systems: the nearest integer GDMS, the Gauss-like continued fraction system, and the Renyi-like continued fraction system. We prove that these systems have full HD spectrum. A special attention is given to the backward continued fraction system that we introduce and we prove that it has full HD spectrum. This system turns out to be a parabolic iterated function system and this makes the analysis more involved. Several examples have been constructed in the past of systems not having full HD spectrum. We give an example of such a system whose limit set has positive Lebesgue measure.

In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems. We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia set whose points escape to infinity, and in many cases we find the exact dimension. While the upper bound was known previously in the autonomous case, the lower bound was not known in this setting, and all of these results are new in the non-autonomous setting. We also use transfer operator techniques to prove an almost sure invariance principle for random dynamical systems for which the thermodynamical formalism has been well established. In particular, we see that if a system exhibits a fiberwise spectral gap property and the base dynamical system is sufficiently well behaved, i.e. it exhibits an exponential decay of correlations, then the almost sure invariance principle holds. We then apply these results to uniformly expanding random systems like those studied by Mayer, Skorulski, and Urbanski and Denker and Gordin.

For a dynamical system on a metric space a shrinking-target set consists of those points whose orbit hit a given ball of shrinking radius infinitely often. Historically such sets originate in Diophantine approximation, in which case they describe the set of well-approximable numbers. One aspect of such sets that is often studied is their Hausdorff dimension. We will show that an analogue of Bowen's dimension formula holds for such sets when they are generated by conformal non-autonomous iterated function systems satisfying some natural assumptions.

This diploma project is focused on use the fractals in diffractive optical elements. Properties of diffraction patterns of various fractals are investigated. Properties of some simpler non-fractal structures are investigated too. Diffraction patterns are created by computer with program written in Delphi programming language. Potentialities of use the fractals in security optical elements (holograms) and their advantages against the classic non-fractal structures.

The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.