Very badly ordered cycles of interval maps

ABSTRACT

We prove that a periodic orbit P with coprime over-rotation pair is an over-twist periodic orbit iff the P-linear map has the over-rotation interval with left endpoint equal to the over-rotation number of P. We show that this fails if the over-rotation pair of P is not coprime. We give examples of patterns with non-coprime over-rotation pairs, no block structure over over-twists, and with over-rotation number equal to the left endpoint of the forced over-rotation interval (call them very badly ordered, similar to patterns of degree one circle maps in [L. Alseda, J. Llibre, and M. Misiurewicz, Badly ordered cycles of circle maps, Pacific J. Math. 184 (1998), pp. 23–41]). This presents a situation in which the results about over-rotation numbers on the interval and those about classical rotation numbers for circle degree one maps are different. In the end, we explicitly describe the strongest unimodal pattern that forces a given over-rotation interval and use it to construct unimodal very badly ordered patterns with arbitrary non-coprime over-rotation pairs.

Keywords:

• Over-rotation pair
• over-rotation number
• pattern
• periodic orbit

2010 Mathematics Subject Classifications:

• primary 37E05
• 37E15
• secondary 37E45

Disclosure statement

No potential conflict of interest was reported by the authors.