Ultrametrics, Banach's fixed point theorem and the Riordan group

Authors:
Ana Luzón;Manuel A. Morón
Affiliations:
Departamento de Matemática Aplicada a los Recursos Naturales, E.T. Superior de Ingenieros de Montes, Universidad Politécnica de Madrid, 28040-Madrid, Spain;Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense de Madrid, 28040- Madrid, Spain
Venue:
Discrete Applied Mathematics
Year:
2008

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Abstract

We interpret the reciprocation process in K[[x]] as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation of certain arithmetical triangles introduced herein. Later we recognize, as a special case of our construction, the so-called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.