The Universal Cardinal Ordering of Fixed Points

TL;DR

This paper introduces a universal theorem that determines the ordering of fixed points in any period doubling cascade, providing a new way to understand symbolic sequences without prior orbit information.

Contribution

The theorem uniquely determines the permutation of fixed points and generates the orbit and symbolic sequence without needing previous orbit data.

Findings

The theorem applies universally to period doubling cascades.

It resolves issues with ambiguous symbolic sequences.

It requires no prior information about other orbits.

Abstract

We present the theorem which determines, by a permutation, the cardinal ordering of fixed points for any orbit of a period doubling cascade. The inverse permutation generates the orbit and the symbolic sequence of the orbit is obtained as a corollary. The problem present in the symbolic sequences is solved. There, repeated symbols appear, for example, the R (right), which cannot be distinguished among them as it is not known which R is the rightmost of them all. Therefore, there is a lack of information about the dynamical system. Interestingly enough, it is important to point that this theorem needs no previous information about any other orbit.