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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.

<p>In this thesis, we study several applications of self-affine sets to wavelet theory. Five major topics are considered here: wavelet sets (scaling sets), multiwavelet sets (generalized scaling sets), self-affine tiles, integral self-affine multi-tiles, self-affine sets. We divide the thesis into six chapters to discuss these topics. In Chapter 1, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup(K+d_2)$, where $B=A^t$ and $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. We show that it must be a constant in dimension $n=1$ or $2$ and it is bounded by $2\lvert K\rvert$ for any $n$. This result shows that all $A$-dilation self-affine scaling sets must be $A$-dilation MRA scaling sets in dimensions one and two. There exist results on the connection between the theory of wavelets and the theory of integral self-affine tiles and in particular, on the construction of wavelet bases using integral self-affine tiles. However, there are many non-integral self-affine tiles which can also yield wavelet basis. In Chapter 2 and Chapter 3, we give a complete characterization of all two dimensional $A$-dilation scaling sets $K$ such that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup (K+d_2)$ for some $d_1, d_2\in\mathbb{R}^2$, where $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. In Chapter 2, we deal with a particular case where $0\in\{d_1,d_2\}$, i.e. a self-affine tile $K$ satisfies $BK=K\bigcup (K+d)$ for some $d\in\mathbb{R}^2$. Chapter 3 is devoted to the general case with $d_1, d_2\in\mathbb{R}^2$. Moreover, we give a sufficient condition for a self-affine tile, possibly non-integral, to be an MRA scaling set in Chapter 3. Gabardo and Yu first considered using integral self-affine tiles in the Fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain self-similarity properties. In Chapter 4, we generalize their results to the integral self-affine multi-tiles setting. We characterize some analytic properties of integral self-affine multi-tiles under certain conditions. We also consider the problem of constructing (multi)wavelet sets using integral self-affine multi-tiles. Suppose that a measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with an $n\times n$ integral expansive matrix $B$. To our knowledge, no one considered how to represent an integral self-affine $\mathbb{Z}^n$-tiling set as the disjoint union of prototiles. In Chapter 5, we provide an algorithm to decompose $K$ into disjoint pieces $K_j$ which satisfy $K=\displaystyle\bigcup K_j$ such that the collection of the sets $K_j$ is an integral self-affine collection associated with matrix $B$ and the number of pieces $K_j$ is minimal. Using this algorithm, we can determine whether a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with any given $n\times n$ integral expansive matrix $B$. Furthermore, the minimal decomposition we provide is unique. Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In Chapter 6, we not only consider the problem how to compute the Lebesgue measure of self-affine sets $K(B,\mathcal{D})$, but also consider the Hausdorff measure for those with zero Lebesgue measure under the assumption that $K(B,\mathcal{D})$ is a self-similar set. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $\mu=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}\delta_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $\mu$.</p> / Doctor of Science (PhD)

The response of geological materials at the scale of meters and the response of buried targets of different shapes and sizes using controlled-source electromagnetic induction (CSEM) is investigated. This dissertation focuses on three topics; i) frac- tal properties on electric conductivity data from near-surface geology and processing techniques for enhancing man-made target responses, ii) non-linear inversion of spa- tiotemporal data using continuation method, and iii) classification of CSEM transient and spatiotemporal data. In the first topic, apparent conductivity profiles and maps were studied to de- termine self-affine properties of the geological noise and the effects of man-made con- ductive metal targets. 2-D Fourier transform and omnidirectional variograms showed that variations in apparent conductivity exhibit self-affinity, corresponding to frac- tional Brownian motion. Self-affinity no longer holds when targets are buried in the near-surface, making feasible the use of spectral methods to determine their pres- ence. The difference between the geology and target responses can be exploited using wavelet decomposition. A series of experiments showed that wavelet filtering is able to separate target responses from the geological background. In the second topic, a continuation-based inversion method approach is adopted, based on path-tracking in model space, to solve the non-linear least squares prob- lem for unexploded ordnance (UXO) data. The model corresponds to a stretched- exponential decay of eddy currents induced in a magnetic spheroid. The fast inversion of actual field multi-receiver CSEM responses of inert, buried ordnance is also shown. Software based on the continuation method could be installed within a multi-receiver CSEM sensor and used for near-real-time UXO decision. In the third topic, unsupervised self-organizing maps (SOM) were adapted for data clustering and classification. The use of self-organizing maps (SOM) for central- loop CSEM transients shows potential capability to perform classification, discrimi- nating background and non-dangerous items (clutter) data from, for instance, unex- ploded ordnance. Implementation of a merge SOM algorithm showed that clustering and classification of spatiotemporal CSEM data is possible. The ability to extract tar- get signals from a background-contaminated pattern is desired to avoid dealing with forward models containing subsurface response or to implement processing algorithm to remove, to some degree, the effects of background response and the target-host interactions.

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'.

Le transport de l’hydrogène à l’utilisateur final est prévu à l’horizon 2020. Il présente les mêmes problèmes que les gaz de ville classiques du point de vue de la rupture dynamique des tuyaux mais le problème se complique encore car une couche barrière est nécessaire de sorte que les petites molécules H2 ne s’échappent pas. Les modes de rupture et leurs origines dans des tuyaux polymères multi couches sont très mal cernés, voire quasiment inconnus. L’objectif de la thèse est donc d’étudier la problématique de tuyaux polymères, à moyen terme multi couches et armés de fibres, ainsi que la rupture dynamique des matériaux composant ces tuyaux. Pour cela,une première partie est consacrée à la conception et à l’optimisation d’un système de chargement de tubes et une seconde à la détermination de la quantité totale de surface créée au passage de la fissure. Afin que la quantité totale de surface de rupture ne dépende pas de l’outil de mesure, il convient de modéliser la rugosité de surface. Une étude statistique, utilisant les outils développés par B.B. Mandelbrot pour les géométries fractales, permettra de savoir dans quelles mesures un modèle auto-affine est convenable. / The transport of hydrogen to the end user is expected in 2020. It presents the same problems as conventional city gas in terms of the dynamic fracture of pipes but the problem is further complicated because of a barrier layer which is necessary to ensure that small H2 molecules do not escape. The failure modes and their origins in polymer multilayer pipes are very poorly understood and almost unknown. The aim of the Ph.D thesis is to study the problematic of polymer gas pipes, multilayer and composite, as well as the dynamic fracture of materials of these pipes. For this, the first part is devoted to the design and optimization of an experimental system of loading pipes. The second part consists to determine post-mortem the total amountof created surface after rapid crack propagation. To prevent the dependence of the amount of fracture surface with the measurement tool, the surface roughness should be modelled. A statistical study of the fracture surface 2 roughness, using the tools developed by B.B. Mandelbrot for fractal geometries, will allow to know when a self-affine model is suitable.

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