Dynamics of typical Baire-1 functions on a compact n-manifold View Full Text


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Article Info

DATE

2019-02-25

AUTHORS

Bruce Hanson, Pamela Pierce, T. H. Steele

ABSTRACT

Let M be a compact n-dimensional manifold with bB1 the set of Baire-1 self-maps of M. For f∈bB1, let Ω(f)={ω(x,f):x∈M} be the collection of ω-limit sets generated by f, and Λ(f)=∪x∈Mω(x,f) be the set of ω-limit points of f. For a typical f∈bB1, we show the following: for any x∈M, 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; for any ε>0, there exists a natural number K such that fk(M)⊂Bε(Λ(f)) whenever k>K. Moreover, f:Λ(f)→Λ(f) is a bijection, and Λ(f) is closed. The Hausdorff dimension of Λ(f) is zero, and the collection of ω-limit sets Ω(f) is closed in the Hausdorff metric space. The function f is not chaotic in the sense of Li–Yorke, nor in the sense of Devaney. The function f is one-to-one, and the m-fold iterate fm is an element of bB1 for all natural numbers m. More... »

PAGES

1-15

References to SciGraph publications

  • 1984-09. Some typical results on bounded Baire 1 functions in ACTA MATHEMATICA HUNGARICA
  • 1992. Dynamics in One Dimension in NONE
  • 2006-06. Continuity and chaos in discrete dynamical systems in AEQUATIONES MATHEMATICAE
  • 1995. The “Spectral” Decomposition for One-Dimensional Maps in DYNAMICS REPORTED
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    http://scigraph.springernature.com/pub.10.1007/s00010-019-00640-1

    DOI

    http://dx.doi.org/10.1007/s00010-019-00640-1

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    https://app.dimensions.ai/details/publication/pub.1112385667


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