The induced contractive maps on the covering spaces

Document Type : Research Articles

Authors

¹ Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

² Department of Pure Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran

Abstract

‎ ‎Let

(\tilde{X}, p)

be the universal covering space of a compact metrizable space

X

‎, ‎which is compact and locally path connected‎.
‎In this paper‎, ‎we show that there exist metrics

d

and

d^{'}

for

X

and

\tilde{X}

‎, ‎respectively‎, ‎such that any contractive map ‎

f : X \to X

induces a contractive map on

\tilde{X}

. ‎As an‎ application‎, ‎it is obtained that every iterated function system(IFS) on the space

X

with attractor

K

‎, ‎induces an IFS on

\tilde{X}

with attractor

\tilde{K},

such that

p (\tilde{K}) = K

‎.

Keywords