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After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times. / Decision Sciences / PhD. (Operations Research)

The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L {q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L {q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures.

We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.

Au cours de ce travail, nous nous intéressons aux limites d'échelle de deux classes de cartes. Dans un premier temps, nous regardons les quadrangulations biparties de genre strictement positif g fixé et, dans un second temps, les quadrangulations planaires à bord dont la longueur du bord est de l'ordre de la racine carrée du nombre de faces. Nous voyons ces objets comme des espaces métriques, en munissant leurs ensembles de sommets de la distance de graphe, convenablement renormalisée. Nous montrons qu'une carte prise uniformément parmi les cartes ayant n faces dans l'une de ces deux classes tend en loi, au moins à extraction près, vers un espace métrique limite aléatoire lorsque n tend vers l'infini. Cette convergence s'entend au sens de la topologie de Gromov--Hausdorff. On dispose de plus des informations suivantes sur l'espace limite que l'on obtient. Dans le premier cas, c'est presque sûrement un espace de dimension de Hausdorff 4 homéomorphe à la surface de genre g. Dans le second cas, c'est presque sûrement un espace de dimension 4 avec une frontière de dimension 2, homéomorphe au disque unité de R^2. Nous montrons en outre que, dans le second cas, si la longueur du bord est un petit~o de la racine carrée du nombre de faces, on obtient la même limite que pour les quadrangulations sans bord, c'est-à-dire la carte brownienne, et l'extraction n'est plus requise. / In this work, we discuss the scaling limits of two particular classes of maps. In a first time, we address bipartite quadrangulations of fixed positive genus g and, in a second time, planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We view these objects as metric spaces by endowing their sets of vertices with the graph metric, suitably rescaled.We show that a map uniformly chosen among the maps having n faces in one of these two classes converges in distribution, at least along some subsequence, toward a limiting random metric space as n tends to infinity. This convergence holds in the sense of the Gromov--Hausdorff topology on compact metric spaces. We moreover have the following information on the limiting space. In the first case, it is almost surely a space of Hausdorff dimension 4 that is homeomorphic to the genus g surface. In the second case, it is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the unit disc of R^2. We also show that in the second case, if the length of the boundary is little-o of the square root of the number of faces, the same convergence holds without extraction and the limit is the same as for quadrangulations without boundary, that is the Brownian map.

Nous étudions les semi-groupes de matrices avec des points de vue variés qui se re-coupent. Le point de vue de la croissance s’avère relié à un point de vue géométrique : nous avons partiellement généralisé aux semi-groupes un théorème de Patterson-Sullivan-Paulin sur les groupes, qui donne l’égalité entre exposant critique et dimension de Hausdorff de l’ensemble limite. Nous obtenons cela dans le cadre général des semi-groupes d’isométries d’un espace Gromov-hyperbolique, et notre preuve nous a permis d’obtenir également d’autres résultats nouveaux. Le point de vue informatique s’avère également relié à la croissance, puisque la notion de semi-groupe fortement automatique, que nous avons introduit, permet de calculer les exposants critiques exactes de semi-groupes de développement en base β. Et ce point de vue donne également beaucoup d’autres informations sur ces semi-groupes. Cette notion de croissance s’avère aussi reliée à des conjectures sur les fractions continues telles que celle de Zaremba. Et c’est en étudiant certains semi-groupes de matrices que nous avons pu démontrer des résultats sur les fractions continues périodiques bornées qui permettent de petites avancées dans la résolution d'une conjecture de McMullen. / We study matrix semigroups with different point of view that overlaps. The growth point of view seems to be related with the geometric point of view : we partially generalize to the semigroups a theorem on groups of Patterson-Sullivan-Paulin, that give the equality between the critical exponent and the Hausdorff dimension of the limit set. We obtain this in the general framework of isometries of a Gromov-hyperbolic space, and our proof give also others new results. The computer science point of view is also related to the growth, since we obtain a way to calculate exact values of critical exponents of somes β-adic development semigroups, from a notion of automatic semigroups that we introduce. Furthermore this point of view give a lot of information on these semigroups. This notion of growth shows to be also related to conjectures on continued fractions like Zaremba’s one. And by studing some matrix semigroups we were able to prove some results on bounded periodic continued fractions, doing a little step in the resolution of a conjecture of McMullen.

<p>Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.</p><p>In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.</p><p>A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.</p>

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

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