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The periodic points and the invariant set of an ϵ-contractive map

✍ Scribed by Changming Ding; S.B. Nadler Jr.

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
732 KB
Volume
15
Category
Article
ISSN
0893-9659

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✦ Synopsis

It

is shown that the invariant set of an c-contractive map f on a compact metric space X is the same as the set of periodic points of f. Furthermore, the set of periodic points of f is finite and, only assuming that X is locally compact, there is at most one periodic point in each component X. The theorems are applied to prove a known fixed-point theorem, a result concerning inverse limits, a result about periodic points of compositions, and a result showing that c-contractive maps on continua are really contraction maps with a change in metric. It is shown that all our results hold for locally contractive maps on compact metric spaces.

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The invariance of domain theorem for countably 1-set-contractive mappings
✍ Juan A. Gatica; Yun-Ho Kim 📂 Article 📅 2010 🏛 Elsevier Science 🌐 English ⚖ 281 KB

The purpose of this paper is to extend the invariance of domain theorem to a large class of countably 1-γ -contractive maps, by using homotopy theory and degree theory for countably 1-γ -contractive maps.