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Horseshoes for continuous mappings of an interval

  • M. Misiurewicz
Chapter
Part of the Progress in Mathematics book series (PM, volume 8)

Abstract

Let X be a compact Hausdorff space and let f: X → X be a continuous mapping. Let us recall the notion of topological entropy. For a cover A of X we put N(A) = min {Card B: B is a subcover of A}, \(h(f,A) = \mathop {\lim }\limits_{n \to \infty } \tfrac{1}{n}\log N({A^n})\), \({A^n} = \{ {a_0} \cap {f^{ - 1}}{a_1} \cap ... \cap {f^{ - (n - 1)}}{a_{n -1}}:{a_0},{a_1},...,{a_{n - 1}} \in A\} \) n (the limit always exists and is not greater than log N(A)). The topological entropy of f is defined as
$$h(f) = \sup \{ h(f,A):A{\kern 1pt} {\kern 1pt} is{\kern 1pt} an{\kern 1pt} {\kern 1pt} open{\kern 1pt} {\kern 1pt} \,\operatorname{cov} er\,of\,X\} $$
.

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References

  1. [1]
    R. Bowen, J. Franks — The periodic points of maps of the disk and the interval — Topology 15 (1976), 337 – 342CrossRefGoogle Scholar
  2. [2]
    T. Li, J. Yorke — Period three implies chaos — Amer. Math. Monthly 82 (1975), 985 – 992CrossRefGoogle Scholar
  3. [3]
    M. Misiurewicz, W. Szlenk — Entropy of piecewise monotone mappings — Astérisque 50 (1977) , 229 – 310 (the full version will appear in Studia Math. 67)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • M. Misiurewicz
    • 1
  1. 1.Institute of MathematicsWarsaw UniversityWarszawaPoland

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