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| A Necessary and sufficient Condition for Finiteness of Nonwandering Sets of Mappings of the Interval |
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Citation： |
Zhou Zuoling.A Necessary and sufficient Condition for Finiteness of Nonwandering Sets of Mappings of the Interval[J].Chinese Annals of Mathematics B,1982,3(1):121~130 |
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Net amount： 798 |
Authors： |
Zhou Zuoling; |
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| Abstract： |
Denote C^0(I, I) the set of all continuous mappings of the interval. Let $f \in c^0(I,I)$, for any positive integer n, we define f^n inductively by f^1=f , f^n=f*f^n-1 and f^0= the identity map.
Definition 1. Let If there exists a positive integer n such that f^n(p) =p, then p is called a periodic point of f.
Denote P(f) the set of all periodic points of f.
Definition 2. Let p \in I. If for any neighborhood U(p) of p, there exists k>0 sue h that f^k(U(p)) \bigcap U(p) \neq ф, then p is called a non wandering point of f.
Denote $\[\Omega (f)\]$ the set of all nonwandering points of f.
Our main aim of this article is to prove the following theorem which, together with Theorem A in [1] , answers a question of Block [2, p. 358].
Main Theorem. $\[\Omega (f)\]$ is finite for $f \in C^0(I,I)$, if and only if P(f) is finite.
As a consequence, we obtain directly that the topological entropy of f is zero, if P(f) is finite. |
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