On backward attractors of interval maps

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F47813059%3A19610%2F21%3AA0000092" target="_blank" >RIV/47813059:19610/21:A0000092 - isvavai.cz</a>

Result on the web

<a href="https://iopscience.iop.org/article/10.1088/1361-6544/ac23b6/" target="_blank" >https://iopscience.iop.org/article/10.1088/1361-6544/ac23b6/</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1088/1361-6544/ac23b6" target="_blank" >10.1088/1361-6544/ac23b6</a>

Result language

angličtina

Original language name

On backward attractors of interval maps

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2021

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Nonlinearity

ISSN

0951-7715

e-ISSN

1361-6544

Volume of the periodical

34

Issue of the periodical within the volume

11

Country of publishing house

GB - UNITED KINGDOM

Number of pages

31

Pages from-to

7415-7445

UT code for WoS article

000698466200001

EID of the result in the Scopus database

2-s2.0-85117688797