Fine boundary properties in complex analysis and discrete potential theory

Departament responsable de la tesi: Departament de Matemàtiques.In this thesis we study some fine properties of sets in the boundary of continuous and discretemetric spaces. On the discrete side, we consider a Potential Theory on infinite trees. Usingprobabilistic methods, we derive a description of the set of irregular points for the Dirichletproblem on the tree. In particular, we obtain a Wiener's type test and we show that the setof irregular points has zero capacity. We also discuss some uniqueness results for the solutionof the Dirichlet problem in some energy spaces. Then, we provide an equilibrium equationcharacterizing measures that realize a p-capacity on the natural boundary of the tree and wediscuss a quite surprising application to the classical problem of tiling a rectangle with squares.In the continuous setting, we study metric distortion properties of sets in unit circle underthe action of inner functions. Classical results by Löowner and by Fernández-Pestana describethis distortion in terms of Lebesgue measure and Hausdorff content respectively, for innerfunctions having the Denjoy-Wolff point in the unit disc. We present an extended theoremof the same kind which applies also to inner functions with no fixed points in the unit disc.In this situation, the distortion properties are given in terms of a natural (infinite) measurewhich provides at the same time information on the size and on the distribution of a set aroundthe Denjoy-Wolff point. As an application of our result we derive an estimate of the size ofthe omitted values of an inner functions in terms of the size of points in the unit circle notadmitting a finite angular derivative. Using our result we are also able to prove a version ofLöwner and Fernández-Pestana theorems for inner functions of the upper half plane fixing thepoint at infinity