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

10.22080/cjms.2020.18710.1487

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$‎.

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