Real Analysis Exchange

Minimal and ω-Minimal Sets of Functions with Connected Gδ Graphs

Michaela Čiklová

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Abstract

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 $\varphi\in\mathcal J$.

Article information

Source
Real Anal. Exchange, Volume 32, Number 2 (2006), 397-408.

Dates
First available in Project Euclid: 3 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1199377480

Mathematical Reviews number (MathSciNet)
MR2369852

Zentralblatt MATH identifier
1116.37004

Subjects
Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]
Secondary: 37B40: Topological entropy 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

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

Citation

Čiklová, Michaela. Minimal and ω-Minimal Sets of Functions with Connected G δ Graphs. Real Anal. Exchange 32 (2006), no. 2, 397--408. https://projecteuclid.org/euclid.rae/1199377480


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