Startseite Dynamics of typical Baire-1 functions on the interval
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Dynamics of typical Baire-1 functions on the interval

  • T. H. Steele EMAIL logo
Veröffentlicht/Copyright: 15. November 2017

Abstract

Let I=[0,1], and let bB1 be the set of Baire-1 self-maps of I. For fbB1, let Λ(f)=xIω(x,f) be the set of ω-limit points of f. We prove the following:

  1. There exists a residual subset S of bB1 such that for any fS and xI the ω-limit set ω(x,f) is contained in the set of points at which f is continuous, and ω(x,f) is an -adic adding machine.

  2. There exists a residual subset S of bB1 such that for any fS and for any ε>0 there exists a natural number M such that fm(I)Bε(Λ(f)) whenever m>M. Moreover, f:Λ(f)Λ(f) is a bijection.

MSC 2010: 54H20; 37B99; 26A18

References

[1] S. J. Agronsky, A. M. Bruckner and M. Laczkovich, Dynamics of typical continuous functions, J. Lond. Math. Soc. (2) 40 (1989), 227–243. 10.1112/jlms/s2-40.2.227Suche in Google Scholar

[2] N. C. Bernardes and U. B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math. 231 (2012), 1655–1680. 10.1016/j.aim.2012.05.024Suche in Google Scholar

[3] L. Block and W. Coppel, Dynamics in one Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1991. 10.1007/BFb0084762Suche in Google Scholar

[4] L. Block and J. Keesling, A characterization of adding machines, Topology Appl. 140 (2004), 151–161. 10.1016/j.topol.2003.07.006Suche in Google Scholar

[5] A. Blokh, The spectral decomposition for one-dimensional maps, Dynamics Reported. Expositions in Dynamical Systems. New series. Vol. 4, Springer, Berlin (1995), 1–59. 10.1007/978-3-642-61215-2_1Suche in Google Scholar

[6] A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice-Hall, Upper Saddle River, 1997. Suche in Google Scholar

[7] A. M. Bruckner and G. Petruska, Some typical results of bounded Baire-1 functions, Acta Math. Hungar. 43 (1984), 325–333. 10.1007/BF01958029Suche in Google Scholar

[8] J. Buescu and I. Stewart, Lyapunov stability and adding machines, Ergodic Theory Dynam. Systems 15 (1995), 271–290. 10.1017/S0143385700008373Suche in Google Scholar

[9] E. D’Aniello, U. Darji and T. H. Steele, Ubiquity of odometers in topological dynamical systems, Topology Appl. 156 (2008), 240–245. 10.1016/j.topol.2008.07.003Suche in Google Scholar

[10] H. Lehning, Dynamics of typical continuous functions, Proc. Amer. Math. Soc. 123 (1995), 1703–1707. 10.1090/S0002-9939-1995-1239798-XSuche in Google Scholar

[11] Z. Nitecki, Topological Dynamics on the Interval, Progr. Math. 21, Birkhäuser, Basel, 1982. 10.1007/978-1-4899-2689-0_1Suche in Google Scholar

[12] T. H. Steele, Continuity and chaos in discrete dynamical systems, Aequationes Math. 71 (2006), 300–310. 10.1007/s00010-005-2813-7Suche in Google Scholar

Received: 2017-3-23
Revised: 2017-9-17
Accepted: 2017-9-19
Published Online: 2017-11-15
Published in Print: 2017-12-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 25.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jaa-2017-0009/html
Button zum nach oben scrollen