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## On the $\omega$-limit sets of tent maps

### Tom 217 / 2012

Fundamenta Mathematicae 217 (2012), 35-54 MSC: 37B10, 37C50, 37E05, 54C05, 54H20. DOI: 10.4064/fm217-1-4

#### Streszczenie

For a continuous map $f$ on a compact metric space $(X,d)$, a set $D\subset X$ is internally chain transitive if for every $x,y\in D$ and every $\delta>0$ there is a sequence of points $\langle x=x_0,x_1,\ldots,x_n=y\rangle$ such that $d(f(x_i),x_{i+1})< \delta$ for $0\leq i< n$. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an $\omega$-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an $\omega$-limit set. Together, these results lead us to conjecture that for tent maps with shadowing, the $\omega$-limit sets are precisely those sets having internal chain transitivity.

#### Autorzy

• Andrew D. BarwellSchool of Mathematics
University of Bristol
Howard House
Queens Avenue
Bristol, BS8 1SN, UK
and
School of Mathematics
University of Birmingham
Birmingham, B15 2TT, UK
e-mail
• Gareth DaviesMathematical Institute
University of Oxford
24-29 St. Giles'
Oxford, OX1 3LB, UK
e-mail
• Chris GoodSchool of Mathematics
University of Birmingham
Birmingham, B15 2TT, UK
e-mail

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