Forcing among patterns with no block structure

TL;DR

This paper investigates the ordering of natural numbers based on cycle structures in continuous interval maps, establishing conditions under which certain cycle periods imply the existence of others without block structure.

Contribution

It introduces a new ordering among natural numbers and proves that certain cycle structures imply the existence of others in continuous interval maps.

Findings

If a cycle of period m exists with no block structure, then a cycle of period s also exists under the order m bb s.

The paper characterizes the possible sets of periods for cycles with no division or block structure.

It provides a framework to understand the relationship between cycle periods in continuous interval maps.

Abstract

Define the following order among all natural numbers except for 2 and 1: \[ 4\gg 6\gg 3\gg \dots \gg 4n\gg 4n+2\gg 2n+1\gg 4n+4\gg\dots \] Let $f$ be a continuous interval map. We show that if $m\gg s$ and $f$ has a cycle with no division (no block structure) of period $m$ then $f$ has also a cycle with no division (no block structure) of period $s$. We describe possible sets of periods of cycles of $f$ with no division and no block structure.