HANTÁKOVÁ, Jana and Samuel Joshua ROTH. On backward attractors of interval maps. Nonlinearity. Bristol (GB): IOP Publishing Ltd, vol. 34, No 11, p. 7415-7445. ISSN 0951-7715. doi:10.1088/1361-6544/ac23b6. 2021.
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Basic information
Original name On backward attractors of interval maps
Authors HANTÁKOVÁ, Jana (203 Czech Republic, guarantor, belonging to the institution) and Samuel Joshua ROTH (840 United States of America, belonging to the institution).
Edition Nonlinearity, Bristol (GB), IOP Publishing Ltd, 2021, 0951-7715.
Other information
Original language English
Type of outcome Article in a journal
Field of Study 10101 Pure mathematics
Country of publisher United Kingdom of Great Britain and Northern Ireland
Confidentiality degree is not subject to a state or trade secret
WWW Nonlinearity
RIV identification code RIV/47813059:19610/21:A0000092
Organization unit Mathematical Institute in Opava
Doi http://dx.doi.org/10.1088/1361-6544/ac23b6
UT WoS 000698466200001
Keywords in English interval map; transitivity; alpha-limit set; special alpha-limit set; beta-limit set; backward attractor
Tags
Tags International impact, Reviewed
Changed by Changed by: Mgr. Aleš Ryšavý, učo 28000. Changed: 29/3/2022 09:36.
Abstract
Special alpha-limit sets (s alpha-limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of s alpha-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong's attracting centre that completely characterizes which interval maps have all s alpha-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh's models of solenoidal and basic omega-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of s alpha-limit sets to the dynamics within them. For example, we show that the isolated points in a s alpha-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the s alpha-limit set is nowhere dense. Moreover, we show that s alpha-limit sets in the interval are always both F-sigma and G(delta) . Finally, since s alpha-limit sets need not be closed, we propose a new notion of beta-limit sets to serve as backward attractors. The beta-limit set of x is the smallest closed set to which all backward orbit branches of x converge, and it coincides with the closure of the s alpha-limit set. At the end of the paper we suggest several new problems about backward attractors.
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