A non-Borel special alpha-limit set in the square

Jackson, Steve; Mance, Bill; Roth, Samuel

doi:10.1017/etds.2021.68

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…All of this, applied to the right hand side of ≤ in (17), shows that (18) D(E(y, j), µ n(j) ) → 0 as j → ∞.…”

Section: Resultsmentioning

confidence: 96%

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Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

Deka¹,

Jackson²,

Kwietniak³

et al. 2022

Preprint

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We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a T -invariant Borel probability measure, but we also consider much more general sets (for example, α-Birkhoff regular sets and the irregular set appearing in multifractal analysis of ergodic averages of a continuous realvalued function). We show that many of these sets are Borel. In fact, all these sets are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is in some cases necessary to obtain Borelness. When these sets are Borel, we use the Borel hierarchy to measure their descriptive complexity. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that for many dynamical systems these sets are properly located at the third level. To demonstrate that the specification property is a sufficient, but not necessary condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is Π 0 3complete.

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“…All of this, applied to the right hand side of ≤ in (17), shows that (18) D(E(y, j), µ n(j) ) → 0 as j → ∞.…”

Section: Resultsmentioning

confidence: 96%

“…Also, sets of interest in dynamics are sometimes complete at levels past the Borel hierarchy. For example, in [17] a dynamical system in the unit square is constructed for which the set of special α-limit points is Π 1 1complete.…”

Section: Introductionmentioning

confidence: 99%

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

Deka¹,

Jackson²,

Kwietniak³

et al. 2022

Preprint

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show abstract

“…Outside the realm of one-dimensional dynamics the situation is even more complicated. It has been shown that sα-limit sets are always analytic, but not necessarily closed or even Borel [17,18]. Therefore it seems prudent to study more closely the properties of sα-limit sets in onedimensional dynamics, and especially in the most important one-dimensional space where Hero's work began, the unit interval.…”

Section: Introductionmentioning

confidence: 99%

On backward attractors of interval maps*

Hantáková

Roth

2021

Nonlinearity

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Special α-limit sets (sα-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 sα-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 sα-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 ω-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sα-limit sets to the dynamics within them. For example, we show that the isolated points in a sα-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 sα-limit set is nowhere dense. Moreover, we show that sα-limit sets in the interval are always both F σ and G δ . Finally, since sα-limit sets need not be closed, we propose a new notion of β-limit sets to serve as backward attractors. The β-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 sα-limit set. At the end of the paper we suggest several new problems about backward attractors.

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“…Outside the realm of one-dimensional dynamics the situation is even more complicated. It has been shown that sα-limit sets are always analytic, but not necessarily Borel [22]. If we denote by SA(f ) (respectively, ω(f )) the union of α-limit sets of all backward branches (respectively, all ω-limit sets) of a map f and by Rec(f ) the set of all recurrent points of f , then Rec(f ) ⊆ SA(f ) ⊆ Rec(f ) ⊆ ω(f ), for every map f on the topological graph (see [34], cf.…”

mentioning

confidence: 99%

On the structure of $ \alpha $-limit sets of backward trajectories for graph maps

Foryś-Krawiec¹,

Hantáková²,

Oprocha³

2022

DCDS

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<p style='text-indent:20px;'>In the paper we study what sets can be obtained as <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit sets are <inline-formula><tex-math id="M4">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets and for all but finitely many points <inline-formula><tex-math id="M5">\begin{document}$ x $\end{document}</tex-math></inline-formula>, we can obtain every <inline-formula><tex-math id="M6">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limits set as the <inline-formula><tex-math id="M7">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of a backward trajectory starting in <inline-formula><tex-math id="M8">\begin{document}$ x $\end{document}</tex-math></inline-formula>. For zero entropy maps, every <inline-formula><tex-math id="M9">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.</p>

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A non-Borel special alpha-limit set in the square

Cited by 4 publications

References 24 publications

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

On backward attractors of interval maps*

On the structure of $ \alpha $-limit sets of backward trajectories for graph maps

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