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Iterates of Borel functions. (English) Zbl 1505.37048

Summary: Let \(\mathcal{B}(\alpha)\) be the set of bounded Borel-\(\alpha\) self-maps of \(I = [0, 1]\), where \(\alpha\) is some countable ordinal. For \(f : I \to I\), let \(\omega(x, f)\) be the \(\omega\)-limit set generated by \(x \in I\), and take \(\mathrm{CR}(f)\) to be the set of chain recurrent points of \(f\). There exists \(\mathcal{T}\) a residual subset of \(\mathcal{B}(\alpha)\) such that for any \(f \in \mathcal{T}\), the following hold:
1.
The \(n\)-fold iterate \(f^n\) is an element of \(\mathcal{B}(\alpha)\), for all natural numbers \(n\).
2.
For any \(x \in I\), the \(\omega\)-limit set \(\omega(x, f)\) is a Cantor set.
3.
For any \(\varepsilon > 0\), there exists a natural number \(M\) such that \(f^m(I) \subset B_\varepsilon(\mathrm{CR}(f))\), whenever \(m > M\).
4.
The Hausdorff dimension satisfies \(\dim_{\mathcal{H}} \overline{\mathrm{CR}(f)} = 0\).
5.
There exists \(\mathcal{R}\), a residual subset of \([0, 1]\), with the property that \(\omega_f : \mathcal{R} \to \mathcal{K}\) given by \(x \longmapsto \omega(x, f)\) is continuous.
6.
The function \(f\) is non-chaotic in the senses of Devaney and Li-Yorke.

MSC:

37E05 Dynamical systems involving maps of the interval
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
39B12 Iteration theory, iterative and composite equations
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[1] Agronsky, S. J.; Bruckner, A. M.; Laczkovich, M., Dynamics of typical continuous functions, J. Lond. Math. Soc., 40, 227-243 (1989) · Zbl 0657.58016
[2] Agronsky, S. J.; Bruckner, A. M.; Pearson, T. L., The structure of ω-limit sets for continuous functions, Real Anal. Exch., 15, 483-510 (1989/1990) · Zbl 0728.26006
[3] Alikhani-Koopaei, A., On the sets of fixed points of bounded Baire one functions, Asian-Eur. J. Math., 12, Article 1950040 pp. (2019) · Zbl 1417.37140
[4] Alikhani-Koopaei, A., On chain recurrent sets of typical bounded Baire one functions, Topol. Appl., 257, 1-10 (2019) · Zbl 1410.26011
[5] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacy, P., On Devaney’s definition of chaos, Am. Math. Mon., 99, 332-334 (1992) · Zbl 0758.58019
[6] Bernardes, N. C.; Darji, U. B., Graph theoretic structure of maps of the Cantor space, Adv. Math., 231, 1655-1680 (2012) · Zbl 1271.37028
[7] Block, L.; Coppel, W., Dynamics in One Dimension, Lecture Notes in Mathematics, vol. 1513 (1991), Springer-Verlag: Springer-Verlag Berlin
[8] Block, L.; Keesling, J., A characterization of adding machines, Topol. Appl., 140, 151-161 (2004) · Zbl 1052.37010
[9] Blokh, A., The spectral decomposition for one-dimensional maps, Dyn. Rep., 4, 1-59 (1995) · Zbl 0828.58009
[10] Blokh, A.; Bruckner, A. M.; Humke, P. D.; Smital, J., The space of ω-limit sets of a continuous map of the interval, Trans. Am. Math. Soc., 348, 1357-1372 (1996) · Zbl 0860.54036
[11] Bruckner, A. M.; Bruckner, J. B.; Thomson, B. S., Real Analysis (1997), Prentice-Hall: Prentice-Hall Upper Saddle River · Zbl 0872.26001
[12] Bruckner, A. M.; Ceder, J., Chaos in terms of the map \(x \to \omega(x, f)\), Pac. J. Math., 156, 63-96 (1992) · Zbl 0728.58020
[13] Bruckner, A. M.; Steele, T. H., The Lipschitz structure of continuous self-maps of generic compact sets, J. Math. Anal. Appl., 118, 798-808 (1994) · Zbl 0820.26001
[14] Buescu, J.; Stewart, I., Lyapunov stability and adding machines, Ergod. Theory Dyn. Syst., 15, 271-290 (1995) · Zbl 0848.54027
[15] D’Aniello, E.; Darji, U., Chaos among self-maps of the Cantor space, J. Math. Anal. Appl., 381, 781-788 (2011) · Zbl 1223.37042
[16] D’Aniello, E.; Darji, U.; Steele, T. H., Ubiquity of odometers in topological dynamical systems, Topol. Appl., 156, 240-245 (2008) · Zbl 1153.37003
[17] D’Aniello, E.; Humke, P.; Steele, T. H., The space of adding machines generated by continuous self-maps of manifolds, Topol. Appl., 157, 954-960 (2010) · Zbl 1191.39022
[18] Downarowicz, T., Survey of odometers and Toeplitz flows, Contemp. Math., 385, 7-37 (2005) · Zbl 1096.37002
[19] Kechris, A., Classical Descriptive Set Theory (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.04002
[20] Kuratowski, K., Topology (Volume 1) (1966), Academic Press · Zbl 0158.40901
[21] Kuratowski, K., Sur une generalisation de la notion d’homeomorphie, Fundam. Math., 22, 206-220 (1934) · Zbl 0009.13201
[22] Lehning, H., Dynamics of typical continuous functions, Proc. Am. Math. Soc., 123, 1703-1707 (1995) · Zbl 0843.58077
[23] Nitecki, Z., Topological Dynamics on the Interval, Progr. Math., vol. 21 (1982), Birkhäuser: Birkhäuser Basel · Zbl 0506.54035
[24] Oxtoby, J., Measure and Category (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0217.09201
[25] Steele, T. H., Continuity and chaos in discrete dynamical systems, Aequ. Math., 71, 300-310 (2006) · Zbl 1092.37023
[26] Steele, T. H., Dynamics of typical Baire-1 functions on the interval, J. Appl. Anal., 23, 59-64 (2017) · Zbl 1379.54034
[27] Steele, T. H., The space of ω-limit sets for Baire-1 functions on the interval, Topol. Appl., 248, 59-63 (2018) · Zbl 1402.54035
[28] Steele, T. H., Dynamics of typical Baire-2 functions on the interval, Topol. Appl., 265, Article 106821 pp. (2019) · Zbl 1428.37041
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