<p>In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers.</p><p>Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f.</p><p>In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions.</p><p>In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.</p>

In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers. Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f. In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions. In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.

La première partie de cette thèse porte sur l’étude de la marche aléatoire sur le tore Td := Rd/Zd définie par une mesure de probabilité SLd(Z). Pour étudier le Théorème Central Limite et la loi du logarithme itéré, nous appliquons la méthode de Gordin qui consiste à se ramener à des martingales. Pour cela, nous utilisons un résultat de Bourgain, Furmann, Lindenstrauss et Mozes nous permettant de résoudre l’équation de Poisson pour des points ayant de bonnes propriétés diophantiennes. Dans la deuxième partie, nous étudions la marche sur Rd\{0} définie par l’action de SLd(R) et nous montrons un résultat de vitesse de convergence dans le théorème de renouvellement de Guivarc’h et Le Page. / The first part of this thesis deals with the random walk on the torus Td := Rd/Zd defined by a robability measure on SLd(Z). To study the Central Limit Theorem and the Law of the Iterated Logarithm, we apply Gordin’s method. To do so, we use a result proved by Bourgain, Furmann, Lindenstrauss and Mozes to solve Poisson’s equation at point’s having good diophantine properties.In the second part, we study the walk on Rd \ {0} defined by the action of SLd(R) and we prove a result about the rate of convergence in Guivarc’h and Le Page’s renewal theorem.