Locally contractive maps on perfect Polish ultrametric spaces. (English) Zbl 1461.37015

George, Francis

Mat. Vesn. 68, No. 4, 233-240 (2016).

Summary: In this paper we present a result concerning locally contractive maps defined on subsets of perfect Polish ultrametric spaces (i.e. separable complete ultrametric spaces). Specifically, we show that a perfect compact ultrametric space cannot be contained in its locally contractive image, a corollary relating this result to minimal dynamical systems, and pose a conjecture for the general Polish ultrametric case.

MSC:

37B02	Dynamics in general topological spaces
37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)

Keywords:

contractive map; Polish space; ultrametric space; R-tree; dynamical system