Locally Contractive Maps on Perfect Polish Ultrametric Spaces

TL;DR

This paper investigates the behavior of locally contractive maps on perfect Polish ultrametric spaces, proving that such maps cannot contain the entire space within their images, with implications for dynamical systems.

Contribution

It establishes a new result about the non-containment of perfect compact ultrametric spaces under locally contractive maps and relates it to minimal dynamical systems.

Findings

Perfect compact ultrametric spaces are not contained in their locally contractive images

The result connects to properties of minimal dynamical systems

Poses a conjecture for the general Polish ultrametric case

Abstract

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.