<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)

Nous nous plaçons dans le cadre de l'analyse statique de programmes, et nous nous intéressons aux propriétés numériques, c'est a dire celles qui concernent les valeurs numériques des variables de programmes. Nous essayons en particulier de déterminer une sur-approximation garantie de l'ensemble de valeurs possibles pour chaque variable numérique utilisée dans le programme à analyser. Cette analyse statique est faite dans le cadre de la théorie de l'interprétation abstraite, théorie présentant un compromis entre les limites théoriques d'indécidabilite et de calculabilite et la précision des résultats obtenus. Nous sommes partis des travaux d'Eric Goubault et Sylvie Putot, que nous avons étendus et généralisés. Notre nouveau domaine abstrait, appelé ensembles affines contraints, combine à la fois l'efficacite de calcul des domaines à base de formes affines et le pouvoir ex- pressif des domaines relationnels classiques tels que les octogones ou les polyèdres. Le nouveau domaine a été implémenté pour mettre en évidence l'intérêt de cette combinaison, ses avantages, ses performances et ses limites par rapport aux autres domaines numériques déjà existants. Le formalisme ainsi que les résultats pra- tiques ont fait l'objet de plusieurs publications [CAV 2009, CAV 2010].