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Transactions of the American Mathematical Society

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A complete classification of the piecewise monotone functions on the interval


Author: Stewart Baldwin
Journal: Trans. Amer. Math. Soc. 319 (1990), 155-178
MSC: Primary 58F08; Secondary 54H20, 58F13
DOI: https://doi.org/10.1090/S0002-9947-1990-0961618-9
MathSciNet review: 961618
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Abstract: We define two functions $ f$ and $ g$ on the unit interval $ [0,1]$ to be strongly conjugate iff there is an order-preserving homeomorphism $ h$ of $ [0,1]$ such that $ g = {h^{ - 1}}fh$ (a minor variation of the more common term "conjugate", in which $ h$ need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.


References [Enhancements On Off] (What's this?)

  • [BC] Louis Block and Ethan M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, preprint. MR 871677 (88c:58032)
  • [CE] Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, 1980. MR 613981 (82j:58078)
  • [G] John Guckenheimer, Bifurcations of dynamical systems, Dynamical Systems, C.I.M.E. Lectures, Progress in Math., Vol. 8, Birkhäuser, 1980, pp. 115-231. MR 589591 (82g:58065)
  • [MSS] M. Metropolis, M. L. Stein, and P. R. Stein, On finite limit sets for transformations of the unit interval, J. Combin. Theory 15 (1973), 25-44. MR 0316636 (47:5183)
  • [MT] J. Milnor and P. Thurston, On iterated maps of the interval. I, II, Princeton Univ. Press, 1977.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0961618-9
Keywords: Topologically conjugate, kneading sequence
Article copyright: © Copyright 1990 American Mathematical Society

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