Forcing relation on patterns of invariant sets and reductions of interval maps

Mai, Jie-Hua; Shao, Song

doi:10.1090/S0002-9947-2010-05103-7

Forcing relation on patterns of invariant sets and reductions of interval maps
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by Jie-Hua Mai and Song Shao PDF

Trans. Amer. Math. Soc. 363 (2011), 2517-2549 Request permission

Abstract:

Patterns of invariant sets of interval maps are the equivalence classes of invariant sets under order-preserving conjugacy. In this paper we study forcing relations on patterns of invariant sets and reductions of interval maps. We show that for any interval map $f$ and any nonempty invariant set $S$ of $f$ there exists a reduction $g$ of $f$ such that $g|_S=f|_S$ and $g$ is a monotonic extension of $f|_S$. By means of reductions of interval maps, we obtain some general results about forcing relations between the patterns of invariant sets of interval maps, which extend known results about forcing relations between patterns of periodic orbits. We also give sufficient conditions for a general pattern to force a given minimal pattern in the sense of Bobok. Moreover, as applications, we give a new and simple proof of the converse of the Sharkovskiĭ Theorem and study fissions of periodic orbits, entropies of patterns, etc.

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