Scrambled sets of continuous maps of 1-dimensional polyhedra

Mai, Jiehua

doi:10.1090/S0002-9947-99-02192-3

Scrambled sets of continuous maps of 1-dimensional polyhedra
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Trans. Amer. Math. Soc. 351 (1999), 353-362 Request permission

Abstract:

Let $K$ be a 1-dimensional simplicial complex in $R^3$ without isolated vertexes, $X = |K|$ be the polyhedron of $K$ with the metric $d_K$ induced by $K$, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $K$ is finite, then the interior of every scrambled set of $f$ in $X$ is empty. We also show that if $K$ is an infinite complex, then there exist continuous maps from $X$ to itself having scrambled sets with nonempty interiors, and if $X = R$ or $R_+$, then there exist $C^\infty$ maps of $X$ with the whole space $X$ being a scrambled set.

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