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# On the ω-limit sets of tent maps

Barwell, Andrew D. and Davies, Gareth and Good, Chris (2012) On the ω-limit sets of tent maps. Fundamenta Mathematicae, 217 (1). pp. 35-54. ISSN 0016-2736

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URL of Published Version: http://dx.doi.org/10.4064/fm217-1-4

Identification Number/DOI: doi:10.4064/fm217-1-4

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 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 maps with $$shadowing$$, the $$\omega$$-limit sets are precisely those sets having internal chain transitivity.

Type of Work: Article 2012 (Publication) Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Mathematics QA Mathematics University of Birmingham Institute of Mathematics, Polish Academy of Sciences 1261 YES

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