Bounded distortion homeomorphisms on ultrametric spaces

It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity as the end spaces of bushy trees). A bounded distortion property is found that characterizes power quasi-symmetric homeomorphisms between such ultrametric spaces that are also pseudo-doubling. Moreover, examples are given showing the extent to which the power quasi-symmetry of homeomorphisms is not captured by the quasiconformal and bi-H\"older conditions for this class of ultrametric spaces.Comment: 20 pages, 1 figure. To appear in Ann. Acad. Sci. Fenn. Mat

E-Connectedness, Finite Approximations, Shape Theory and Coarse Graining in Hyperspaces

We use upper semifinite hyperspaces of compacta to describe "-connectedness and to compute homology from finite approximations. We find another connection between "- connectedness and the so called Shape Theory. We construct a geodesically complete R-tree, by means of "-components at different resolutions, whose behavior at infinite captures the topological structure of the space of components of a given compact metric space. We also construct inverse sequences of finite spaces using internal finite approximations of compact metric spaces. These sequences can be converted into inverse sequences of polyhedra and simplicial maps by means of what we call the Alexandroff-McCord correspondence. This correspondence allows us to relate upper semifinite hyperspaces of finite approximation with the Vietoris-Rips complexes of such approximations at different resolutions. Two motivating examples are included in the introduction. We propose this procedure as a different mathematical foundation for problems on data analysis. This process is intrinsically related to the methodology of shape theory. Finally this paper reinforces Robins’s idea of using methods from shape theory to compute homology from finite approximations

Complementary Riordan arrays

Abstract Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices

Ultrametric spaces, valued and semivalued groups arising from the theory of shape

In this paper we construct many generalizad ultrametrics in the sets of shape morphisms between topological spaes. We recognize a topology in these sets which is independen on the shape representation of the spaces. We construct valuations an semivaluations on groups of shape equivalences and on n-th shape groups. We also connect shape theory, for arbitrary topological spaces, with the algebraic theory of generalized ultrametric spaces developed by S. Priess-Crampe and P. Ribenboim among other authors

El problema de la intersección de ANR’s de Borsuk y métricas en el hiperespacio del cubo de Hilbert

En este artículo relacionamos dos problemas abiertos en Homotopía y Teoría de la Forma planteados por Borsuk. Demostramos que la respuesta a, al menos,uno de ellos es negativa, y obtenemos algunas consecuencias

Ultrametrics on Čech homology groups

This paper is devoted to introducing additional structure on Čech homology groups. First, we redefine the Čech homology groups in terms of what we have called approximative homology by using approximative sequences of cycles, just as Borsuk introduced shape groups using approximative maps. From this point on, we are able to construct complete ultrametrics on Čech homology groups. The uniform type (and then the group topology) generated by the ultrametric leads to a shape invariant which we use to deduce topological consequences