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

MathSciNet review:
961618

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Abstract: We define two functions and on the unit interval to be *strongly conjugate* iff there is an order-preserving homeomorphism of such that (a minor variation of the more common term "conjugate", in which 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.

**[BC]**Louis Block and Ethan M. Coven,*Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval*, Trans. Amer. Math. Soc.**300**(1987), no. 1, 297–306. MR**871677**, 10.1090/S0002-9947-1987-0871677-X**[CE]**Pierre Collet and Jean-Pierre Eckmann,*Iterated maps on the interval as dynamical systems*, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR**613981****[G]**John Guckenheimer,*Bifurcations of dynamical systems*, Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978) Progr. Math., vol. 8, Birkhäuser, Boston, Mass., 1980, pp. 115–231. MR**589591****[MSS]**N. Metropolis, M. L. Stein, and P. R. Stein,*On finite limit sets for transformations on the unit interval*, J. Combinatorial Theory Ser. A**15**(1973), 25–44. MR**0316636****[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