Systems of iterative functional equations : theory and applications

Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015We formulate a general theoretical framework for systems of iterative functional equations between general spaces. We find general necessary conditions for the existence of solutions such as compatibility conditions (essential hypotheses to ensure problems are well-defined). For topological spaces we characterize continuity of solutions; for metric spaces we find sufficient conditions for existence and uniqueness. For a number of systems we construct explicit formulae for the solution, including affine and other general non-linear cases. We provide an extended list of examples. We construct, as a particular case, an explicit formula for the fractal interpolation functions with variable parameters. Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. When conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties. We provide a general construction for finding solutions, including an explicit formula showing how, in certain cases, a solution can be constructively determined. We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description/enumeration of periodic and pre-periodic points and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of combinatorial objects such as necklaces and words. We provide some examples. We show the relevance of this for the representation of rational numbers. There are many possible proofs of Fermat's little theorem. We exemplify those using necklaces and dynamical systems. Both methods lead to generalizations. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps. The representation of numbers plays an important role in much of this work. Starting from the classical base p representation we present other type of representation of numbers: signed base p representation, Q-representation and finite base p representation of rationals. There is an extended p representation that generalizes some of the listed representations. We consider the concept of bold play in gambling, where the game has a unique win pay-off. The probability that a gambler reaches his goal using the bold play strategy is the solution of a functional equation. We compare with the timid play strategy and extend to the game with multiple pay-offs.Centro de Matemática e Aplicações Fundamentais da ULisboa; Fundação da Faculdade de Ciências da ULisbo