Exterior discrete semi-flows

An exterior discrete semi-flow is a discrete semi-flow generated by an exterior continuous map. The aim of this doctoral thesis is to profoundly study the notion of exterior discrete semi-flow and to apply it to the analysis of iterative processes induced by some numerical methods. Firstly, we establish connections in dimension 0 among three of the most usual end sets of an exterior space and their analogues in the category of exterior discrete semi-flows: those of Brown-Grossman, Steenrod and Borsuk-Cech. After that, with a view to make a study about basins of attraction of fixed and, in general, m-periodic points associated with a rational map defined on the surface of the sphere S2, we consider externologies given by families of the open subsets that contain determined right-invariant subsets; in this way, we are connecting some novel working purely topological techniques to several notions regarding discrete semi-flows and dynamical systems. Next, in order to set a theoretical framework in which the algorithms that have been designed and implemented along this work make sense, we focus on the development of the study of discrete semi-flows on metric spaces and on the creation of a new procedure to introduce measures on CW-complexes which allow us to estimate in a computational manner and compare the sizes of basins of end points. With this, we are building a bridge between the topological and the computational part of this thesis. Finally, relying on the theorems proved and on the geometry and the complex structure of the Riemann sphere, we present new computer programs written in Sage and Mathematica for visualizing the basins of attraction associated with the end points of a discrete semi-flow induced by a rational function different from the identity defined on the surface of the sphere S2 and for computing the measure of these basins of attraction, up to a certain precision