Barwell, Andrew D. and Davies, Gareth and Good, Chris (2012) On the ωlimit sets of tent maps. Fundamenta Mathematicae, 217 (1). pp. 3554. ISSN 00162736
 URL of Published Version: http://dx.doi.org/10.4064/fm21714 Identification Number/DOI: doi:10.4064/fm21714 For a continuous map \(f\) on a compact metric space $(X,d)$, a set D C X is \(internally\) \(chain\) \(transitive\) if for every x,y \(\epsilon\) D and every \(\sigma\)0> O there is a sequence of points x=x\(_0\),x\(_1\),.....,x\(_n\)=y such that d(f(x\(_i\)) \(_,\)x\(_i\)\(_+\)\(_1\))<\(\sigma\)for 0\(\leq\) i < n. It is known that every \(\omega\)limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D C X is internally chain transitive if and only if D is an \(\omega\)limit set for some point in X, and that the same is also true for the full tent map T\(_2\):[0,1]\(\rightarrow\)[0,1]. 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 nonrecurrent 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 maps with \(shadowing\), the \(\omega\)limit sets are precisely those sets having internal chain transitivity. 
Type of Work:  Article 

Date:  2012 (Publication) 
School/Faculty:  Colleges (2008 onwards) > College of Engineering & Physical Sciences 
Department:  School of Mathematics 
Subjects:  QA Mathematics 
Institution:  University of Birmingham 
Copyright Holders:  Institute of Mathematics, Polish Academy of Sciences 
ID Code:  1261 
Refereed:  YES 
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