# Very badly ordered cycles of interval maps

## TL;DR

This paper investigates the properties of over-rotation intervals in interval maps, especially focusing on cases with non-coprime over-rotation pairs, revealing differences from classical rotation number theory and constructing specific complex patterns.

## Contribution

It establishes conditions under which over-twist periodic orbits correspond to over-rotation intervals and introduces very badly ordered patterns with non-coprime over-rotation pairs.

## Key findings

- Over-twist periodic orbit characterization for coprime pairs
- Existence of very badly ordered patterns with non-coprime pairs
- Differences between interval over-rotation and circle rotation theories

## 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 then show that this result fails if the over-rotation pair of $P$ is not coprime. Examples of patterns with non-coprime over-rotation pairs are given so that these patterns have no block structure over over-twists but have over-rotation number equal to the left endpoint of the forced over-rotation interval (such patterns are called \emph{very badly ordered}). 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 elucidate a rigorous description of the strongest unimodal pattern that corresponds to a given over-rotation interval and use it to construct unimodal very badly ordered patterns with arbitrary non-coprime over-rotation pair.