Comparing Periodic Orbits of Maps of the Interval

Bernhardt, Chris; Coven, Ethan M.; Misiurewicz, Michał; Mulvey, Irene

doi:10.2307/2154055

Cited by 9 publications

(3 citation statements)

References 7 publications

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“…Then one obtains a transitive relation → on C. The relation → is a refinement of the Sharkovskiȋ ordering , which has been studied by many authors (see [1,2,3,4,5,6,7,8,9,10,12,23,24], etc.). In the following we will also study this relation.…”

Section: Some Results On Periodic Patternsmentioning

confidence: 99%

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

Mai

Shao

2010

Trans. Amer. Math. Soc.

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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 Sharkovskiȋ Theorem and study fissions of periodic orbits, entropies of patterns, etc.

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Section: Some Results On Periodic Patternsmentioning

confidence: 99%

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

Mai

Shao

2010

Trans. Amer. Math. Soc.

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“…Jungreis [6] provided a combinatorial method to determine if one cycle forces another in certain cases. In [3] a geometric version of Jungreis's algorithm is given and in [4] this algorithm is generalized to any two cycles. In [8], another geometric algorithm is given to determine the forcing relation.…”

Section: Definitions Throughout This Note F : I → I Denotes a Contimentioning

confidence: 99%

Multimodal cycles with linear map having exactly one fixed point

Mulvey

2001

International Journal of Mathematics and Mathematical Sciences

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“…In [3] it is shown that if α is a double then at least one element of I(α) is a representative of α/2 -namely, the tightest. Given the construction of I(α), it is clear that at most one element of I(α) can be a representative of α/2.…”

Section: Lemma 5 If X Is In the K Th Lap Of H J Then The First J Tmentioning

confidence: 99%