Other formats:
BibTeX
LaTeX
RIS
@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.
|