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.

Keywords: contractive map; Polish space; ultrametric space, R-tree; dynamical system.

MSC: 37B05, 54H20, 54E40

Pages:  233$-$240     

Volume  68 ,  Issue  4 ,  2016