Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.

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.

Les recherches présentées dans ce mémoire s’inscrivent dans le cadre des réseaux decapteurs périodiques. Elles portent sur l’étude et la mise en oeuvre d’algorithmes et de protocolesdistribués dédiés à la gestion de données volumineuses, en particulier : la collecte, l’agrégation etla fouille de données. L’approche de la collecte de données permet à chaque noeud d’adapter sontaux d’échantillonnage à l’évolution dynamique de l’environnement. Par ce modèle le suréchantillonnageest réduit et par conséquent la quantité d’énergie consommée. Elle est basée surl’étude de la dépendance de la variance de mesures captées pendant une même période voirpendant plusieurs périodes différentes. Ensuite, pour sauvegarder plus de l’énergie, un modèled’adpatation de vitesse de collecte de données est étudié. Ce modèle est basé sur les courbes debézier en tenant compte des exigences des applications. Dans un second lieu, nous étudions unetechnique pour la réduction de la taille de données massive qui est l’agrégation de données. Lebut est d’identifier tous les noeuds voisins qui génèrent des séries de données similaires. Cetteméthode est basée sur les fonctions de similarité entre les ensembles de mesures et un modèle defiltrage par fréquence. La troisième partie est consacrée à la fouille de données. Nous proposonsune adaptation de l’approche k-means clustering pour classifier les données en clusters similaires,d’une manière à l’appliquer juste sur les préfixes des séries de mesures au lieu de l’appliquer auxséries complètes. Enfin, toutes les approches proposées ont fait l’objet d’études de performancesapprofondies au travers de simulation (OMNeT++) et comparées aux approches existantes dans lalittérature. / This thesis proposes novel big data management techniques for periodic sensor networksembracing the limitations imposed by wsn and the nature of sensor data. First, we proposed anadaptive sampling approach for periodic data collection allowing each sensor node to adapt itssampling rates to the physical changing dynamics. It is based on the dependence of conditionalvariance of measurements over time. Then, we propose a multiple level activity model that usesbehavioral functions modeled by modified Bezier curves to define application classes and allowfor sampling adaptive rate. Moving forward, we shift gears to address the periodic dataaggregation on the level of sensor node data. For this purpose, we introduced two tree-based bilevelperiodic data aggregation techniques for periodic sensor networks. The first one look on aperiodic basis at each data measured at the first tier then, clean it periodically while conservingthe number of occurrences of each measure captured. Secondly, data aggregation is performedbetween groups of nodes on the level of the aggregator while preserving the quality of theinformation. We proposed a new data aggregation approach aiming to identify near duplicatenodes that generate similar sets of collected data in periodic applications. We suggested the prefixfiltering approach to optimize the computation of similarity values and we defined a new filteringtechnique based on the quality of information to overcome the data latency challenge. Last butnot least, we propose a new data mining method depending on the existing K-means clusteringalgorithm to mine the aggregated data and overcome the high computational cost. We developeda new multilevel optimized version of « k-means » based on prefix filtering technique. At the end,all the proposed approaches for data management in periodic sensor networks are validatedthrough simulation results based on real data generated by periodic wireless sensor network.