bibkey: krishnaswamirameshan2026rotor
authors: Govind S. Krishnaswami; Anirudh Rameshan
year: 2026
title: "Symbol sequences from three-rotor coincidences and their word-complexity"
doi: 10.48550/arXiv.2606.25766
url: https://arxiv.org/abs/2606.25766v1
claim: "The paper encodes classical three-rotor trajectories by rotor-coincidence symbol sequences over a four-letter alphabet, proves word-complexity n+3 for ultra-high-energy quasiperiodic orbits through a correspondence with Sturmian cutting sequences, and conjectures that the unique right-special (n-1)-word is the unique n-word starting with 0 read backwards without the 0."
strata_touched:
  - D5/S1/Words/Mechanical/RotorCoincidenceRightSpecial
license: citation-only
triage: anchor

Symbol sequences from three-rotor coincidences and their word-complexity

G. S. Krishnaswami and A. Rameshan, arXiv:2606.25766v1 [nlin.CD]. Quotations are from the v1 TeX source.

Right-special words:

Now, a subword $w$ of $s(\text{g})$ is called right-special if there are at least two symbols $a,b$ such that both $wa$ and $wb$ are subwords of $s(\text{g})$ (\S 1.1 of \cite{fogg-book-2002}).

The conjecture:

\textbf{Conjecture on right-special word.} Based on a study of the examples $\text{sig}=\sqrt{2}$ and $\sqrt{3}$, we conjecture that the $(n-1)$-word that branches (in two ways) coincides with the unique $n$-word starting with 0 read backwards (excluding the 0). For example, when $\text{sig}=\sqrt{2}$, the 6-letter word 021232 tells us that the 5-letter word that branches is 23212 (see Table~\ref{t:Cn-sqrt2}). We have verified this conjecture for $n\le 15$ for $\text{sig}=\sqrt{2}$.

The cutting sequence and the rotor correspondence:

Of relevance to us is the case $\rho =0$ so that the trajectory may be regarded as starting at the origin $(0,0)$. Leaving aside the initial point, we record symbols 1 and 3 when the trajectory subsequently cuts the horizontal and vertical grid lines.

Given a Sturmian sequence $s_1{s}_2{s}_3\cdots$, the corresponding rotor sequence is $02s_{1}2s_{2}2s_{3}2\cdots$.

Open question (1) of the Discussion:

(1) Can we establish/falsify our conjecture from \S \ref{s:conjectures-uhe-quasiperiodic} on the unique right-special $n$-word (that branches in two ways) for symbol sequences of UHE quasiperiodic orbits?

Verified locator

Source: https://arxiv.org/abs/2606.25766v1

• URL: https://arxiv.org/abs/2606.25766v1 (source 3rotor-symb-dyn.tex, md5 5d78bd4b19257909d542bbe5b1469202):
  • right-special words, l. 697;
  • the conjecture, l. 701;
  • the cutting sequence and the rotor correspondence, l. 831–835;
  • open question (1), l. 917.
• DOI: 10.48550/arXiv.2606.25766 (https://doi.org/10.48550/arXiv.2606.25766; arXiv v1 is the only version).