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@article{46707,
author = {Hantáková, Jana and Roth, Samuel Joshua},
article_location = {Bristol},
article_number = {11},
doi = {http://dx.doi.org/10.1088/1361-6544/ac23b6},
keywords = {interval map; transitivity; alpha-limit set; special alpha-limit set; beta-limit set; backward attractor},
language = {eng},
issn = {0951-7715},
journal = {Nonlinearity},
title = {On backward attractors of interval maps},
url = {https://iopscience.iop.org/article/10.1088/1361-6544/ac23b6/},
volume = {34},
year = {2021}
}
TY - JOUR
ID - 46707
AU - Hantáková, Jana - Roth, Samuel Joshua
PY - 2021
TI - On backward attractors of interval maps
JF - Nonlinearity
VL - 34
IS - 11
SP - 7415-7445
EP - 7415-7445
PB - IOP Publishing Ltd
SN - 0951-7715
KW - interval map
KW - transitivity
KW - alpha-limit set
KW - special alpha-limit set
KW - beta-limit set
KW - backward attractor
UR - https://iopscience.iop.org/article/10.1088/1361-6544/ac23b6/
N2 - 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.
ER -
HANTÁKOVÁ, Jana and Samuel Joshua ROTH. On backward attractors of interval maps. \textit{Nonlinearity}. Bristol: IOP Publishing Ltd, 2021, vol.~34, No~11, p.~7415-7445. ISSN~0951-7715. Available from: https://dx.doi.org/10.1088/1361-6544/ac23b6.
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