Chain transitive induced interval maps on continua

Mykola Matviichuk; Damoon Robatian

doi:10.3934/dcds.2015.35.741

Article Contents

2015, Volume 35, Issue 2: 741-755. Doi: 10.3934/dcds.2015.35.741

This issue Previous Article Localization of mixing property via Furstenberg families Next Article An ergodic theory approach to chaos

Chain transitive induced interval maps on continua

Mykola Matviichuk^1, and
Damoon Robatian^2,

1.
University of Toronto, Bahen Centre 40 St. George St., Room 6290, Toronto, ON, M5S 2E4
2.
Institute of Mathematics, NASU, Tereshchenkivs'ka 3, 01601, Kyiv

Received: September 30, 2013

Revised: February 28, 2014

Published: January 2015

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Abstract

Let $f:I\rightarrow I$ be a continuous map of a compact interval $I$ and $C(I)$ be the space of all compact subintervals of $I$ with the Hausdorff metric. We investigate chain transitivity of induced maps on subcontinua of $C(I)$. In particular, we prove the following theorem: Let $\mathcal{M}$ be a subcontinuum of $C(I)$ having at most countably many partitioning points. Then, the induced map $\mathcal{F}:C(I)\to C(I)$ $($i.e. $\mathcal{F}(A):=\{f(x):x\in A\}$ for each $A \in C(I)$$)$ is chain transitive on $\mathcal{M}$ iff $\mathcal{F}^{2}\vert_{\mathcal{M}}=Id$.

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Mathematics Subject Classification: Primary: 37B20; Secondary: 37B45.

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