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The transitivity of induced maps. (English) Zbl 1234.54017
Summary: For a metric continuum $X$, we consider the hyperspaces ${2}^{X}$ and $C\left(X\right)$ of closed and nonempty subsets of $X$ and of subcontinua of $X$, respectively, both with the Hausdorff metric. For a given map $f:X\to X$ we investigate the transitivity of the induced maps ${2}^{f}:{2}^{X}\to {2}^{X}$ and $C\left(f\right):C\left(X\right)\to C\left(X\right)$. Among other results, we show that if $X$ is a dendrite or a continuum of type $\lambda$ and $f:X\to X$ is a map, then $C\left(f\right)$ is not transitive. However, if $X$ is the Hilbert cube, then there exists a transitive map $f:X\to X$ such that ${2}^{f}$ and $C\left(f\right)$ are transitive.
##### MSC:
 54B20 Hyperspaces (general topology) 37B45 Continua theory in dynamics 54F50 Spaces of dimension $\le 1$; curves, dendrites 37B40 Topological entropy