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Kolyada–Misiurewicz–Snoha conjecture on closed special alpha-limit sets

Let f:[0,1]→[0,1]f:[0,1]\to[0,1] be continuous. A special α\alpha-limit set sα(x)s\alpha(x) is the union of the accumulation-point sets of all backward orbit branches of xx. Kolyada–Misiurewicz–Snoha's conjecture. Every special α\alpha-limit set of ff is closed. The paper disproves this conjecture by constructing an interval map with a non-closed special α\alpha-limit set.

Equivalent formulations 1

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  1. Kolyada–Misiurewicz–Snoha conjecture on closed special alpha-limit sets

    Let f:I→If:I\to I be a continuous map of a compact interval, and let x∈Ix\in I. The Kolyada–Misiurewicz–Snoha conjecture. The special α\alpha-limit set sαf(x)s\alpha_f(x) is closed.

    source: Roberto De Leo, “Backward asymptotics in S-unimodal maps”, arXiv:2109.02759 (2022).

Sources & referencesView supporting material

Primary source

Jana Hantáková and Samuel Roth, “On backward attractors of interval maps”, arXiv:2007.10883 (2020).

alpha-limit-setsdynamical-systemsinterval-dynamicsinterval-maps

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