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2006/2007 Minimal and ω-Minimal Sets of Functions with Connected Gδ Graphs
Michaela Čiklová
Real Anal. Exchange 32(2): 397-408 (2006/2007).


Let $I=[0,1]$, and let $\mathcal J$ be the class of functions $I \rightarrow I$ with connected $G_{\delta}$ graph. Recently it was shown that dynamical systems generated by maps in $\mathcal J$ have some nice properties. Thus, the Sharkovsky's theorem is true, and a map has zero topological entropy if and only if every periodic point has period $2^{n}$, for an integer $n\ge 0$. In this paper we consider, for a map $\varphi$ in $\mathcal J$, properties of $\omega$-minimal sets; i.e., sets $M\subset I$ such that the $\omega$-limit set $\omega_{\varphi}(x)$ is $M$, for every $x \in M$. If $\varphi $ is continuous, then, as is well-known, $M$ is minimal if and only if $M$ is non-empty, closed, $\varphi(M)\subseteq M$, any point in $M$ is uniformly recurrent, and no proper subset of $M$ has this property. In this paper we prove that the same is true for $\varphi\in\mathcal J$ with zero topological entropy, but not for an arbitrary $\varphi\in\mathcal J$.


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Michaela Čiklová. "Minimal and ω-Minimal Sets of Functions with Connected Gδ Graphs." Real Anal. Exchange 32 (2) 397 - 408, 2006/2007.


Published: 2006/2007
First available in Project Euclid: 3 January 2008

zbMATH: 1116.37004
MathSciNet: MR2369852

Primary: 26A18 , 37E05 , 54H20‎
Secondary: 26A21 , 37B40

Keywords: $\omega$-minimal set , dynamics of weakly discontinuous map , maps with connected graph , minimal set , topological entropy

Rights: Copyright © 2006 Michigan State University Press

Vol.32 • No. 2 • 2006/2007
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