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arxiv: 2604.08339 · v1 · submitted 2026-04-09 · 🧮 math.DS

Words and numbers: a dynamical systems perspective

Stefano Isola , Francesco Marchionni This is my paper

Pith reviewed 2026-05-10 17:30 UTC · model grok-4.3

classification 🧮 math.DS
keywords Stern-Brocot treeFarey-Christoffel wordsSturmian sequencessubstitution rulesdynamical systemshyperbolic geometrygeodesicsFord circles
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The pith

Substitution rules on Farey-Christoffel words exactly parallel the maps generating permutations of the Stern-Brocot tree and establish a complete correspondence with geodesic and horocyclic motions in the upper half-plane.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper works to unify the ordering of rational numbers with the combinatorics of certain words through dynamical systems. It builds substitution rules for Farey-Christoffel words that mirror the vertical and horizontal maps on positive reals used to permute the Stern-Brocot tree. These rules also produce a natural self-map on the space of all infinite Sturmian sequences. The central result identifies motions up and down the tree with specific flows along scattering geodesics and around Ford circles in the hyperbolic plane. Readers would care if this lets them translate questions about rational approximations into questions about word substitutions and geometric paths.

Core claim

A set of substitution rules is constructed that act on Farey-Christoffel words in exact parallel to the maps on positive reals that generate the permuted Stern-Brocot tree, both vertically and horizontally. These rules induce a map of the space of infinite Sturmian sequences into itself. A complete correspondence is obtained between the vertical and horizontal motions on the SB tree and the geodesic motions along scattering geodesics and the horocyclic motion along Ford circles in the upper half-plane.

What carries the argument

Substitution rules on Farey-Christoffel words that act in parallel to the maps on positive reals generating the permuted Stern-Brocot tree, thereby linking word combinatorics to hyperbolic geometry.

Load-bearing premise

The newly constructed substitution rules on Farey-Christoffel words act in exact parallel to the maps on positive reals that generate the permuted Stern-Brocot tree, without additional constraints or exceptions.

What would settle it

A concrete Farey-Christoffel word for which the substitution rule produces a sequence that does not correspond to the image rational under the parallel map on the Stern-Brocot tree, or a tree motion whose geometric image fails to follow a scattering geodesic or Ford-circle horocycle.

Figures

Figures reproduced from arXiv: 2604.08339 by Francesco Marchionni, Stefano Isola.

Figure 1
Figure 1. Figure 1: The first five levels of the Stern-Brocot tree [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The first four level of the Farey-Christoffel words tree [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 10
Figure 10. Figure 10: On the left, movement on the second level of [PITH_FULL_IMAGE:figures/full_fig_p039_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Movement on the third level with [h + 2 ] (the descent) followed by h − 1 , then h + 2 and lastly h − 1 39 [PITH_FULL_IMAGE:figures/full_fig_p039_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Transition to the fourth level with h + −2 followed by h − 3 [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Movement on the fourth level with [h − 3 ] (the descent) followed by h + −1 , then h − 3 , then h + −1 , then h − 5 , then h + −1 , then h − 3 , and lastly h + −1 40 [PITH_FULL_IMAGE:figures/full_fig_p040_13.png] view at source ↗
read the original abstract

Along with some known and less known results, we discuss new insights relating combinatorics of words and the ordering of the rationals from a dynamical systems point of view, somehow continuing along the path started in [BI]. We obtain in particular a set of results that structure and enrich the correspondence between the Stern-Brocot (SB) ordering of rational numbers and the corresponding ordering of Farey-Christoffel (FC) words, a class of words that, since their appearance in literature at the end of the 18th century, have revealed numerous relationships with other fields of mathematics. Among the results obtained here is the construction of substitution rules that act on the FC words in a parallel way to the maps on the positive reals that generate the permuted SB tree both vertically and horizontally. We further show that these rules naturally induce a map of the space of (infinite) Sturmian sequences into itself. Finally, a complete correspondence is obtained between the vertical and horizontal motions on the SB tree and the geodesic motions along scattering geodesics and the horocyclic motion along Ford circles in the upper half-plane, respectively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript explores connections between the combinatorics of Farey-Christoffel (FC) words and the Stern-Brocot (SB) ordering of positive rationals from a dynamical systems viewpoint, continuing prior work [BI]. It constructs substitution rules on FC words that parallel the vertical and horizontal generating maps on the permuted SB tree, shows that these induce a self-map on the space of infinite Sturmian sequences, and claims a complete correspondence between SB-tree motions and geodesic flows along scattering geodesics together with horocyclic flows along Ford circles in the upper half-plane.

Significance. If the asserted exact parallelism of the substitution rules holds uniformly and the induced dynamical equivalence is complete, the paper would supply a coherent dynamical-systems bridge between word combinatorics, rational orderings, and hyperbolic geometry. This could enrich the study of Sturmian sequences by furnishing explicit geometric realizations of their substitutions. The work structures and enriches known material rather than introducing parameter-free derivations or machine-checked proofs.

major comments (2)
  1. [Abstract] Abstract: the claim of a 'complete correspondence' between vertical/horizontal SB-tree motions and geodesic/horocyclic flows rests on the construction of substitution rules asserted to act 'in a parallel way' to the maps on positive reals, yet no derivation, domain statement, or verification that the rules commute with mediant/continued-fraction operations for all infinite Sturmian words (including eventually periodic and boundary cases) is supplied.
  2. The weakest assumption flagged in the manuscript—that the new FC-word substitution rules act without additional constraints or exceptions—directly affects the load-bearing claim of completeness; if exceptions exist even on a dense set of infinite words (e.g., at Farey-interval endpoints), the induced self-map on Sturmian sequences cannot be a full dynamical equivalence.
minor comments (1)
  1. [Abstract] The abstract refers to 'some known and less known results' without indicating which statements are novel versus reorganizations of material from [BI].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the need to strengthen the support for our claims of parallelism and completeness. We address both major comments below by committing to explicit derivations, domain statements, and verifications in the revised manuscript. These additions will make the dynamical equivalence fully rigorous without altering the core results.

read point-by-point responses

  1. Referee: [Abstract] the claim of a 'complete correspondence' between vertical/horizontal SB-tree motions and geodesic/horocyclic flows rests on the construction of substitution rules asserted to act 'in a parallel way' to the maps on positive reals, yet no derivation, domain statement, or verification that the rules commute with mediant/continued-fraction operations for all infinite Sturmian words (including eventually periodic and boundary cases) is supplied.

    Authors: We agree that the abstract phrasing would be more precise with explicit supporting material. In the revision we will insert a new subsection (likely in Section 3) that derives the substitution rules directly from the combinatorial definition of Farey-Christoffel words via their continued-fraction expansions. We will state the precise domain: the rules extend to all infinite Sturmian sequences by uniform continuity on the symbolic space. Commutation with mediant operations and continued-fraction shifts will be verified first for finite words, then lifted to eventually periodic sequences (corresponding to rational endpoints) and to the boundary rays of Farey intervals. These checks will be presented as lemmas with short proofs, confirming that the induced self-map on Sturmian sequences is indeed a dynamical equivalence without exceptions. revision: yes

  2. Referee: The weakest assumption flagged in the manuscript—that the new FC-word substitution rules act without additional constraints or exceptions—directly affects the load-bearing claim of completeness; if exceptions exist even on a dense set of infinite words (e.g., at Farey-interval endpoints), the induced self-map on Sturmian sequences cannot be a full dynamical equivalence.

    Authors: The construction in the manuscript ensures uniformity by design: each substitution is defined via the same mediant-based recursion that generates the permuted Stern-Brocot tree, and the correspondence with geodesic/horocyclic flows follows from the identification of FC words with scattering geodesics and Ford circles. Nevertheless, to remove any doubt we will add an explicit check that the rules remain well-defined and commute at all Farey-interval endpoints (including the eventually periodic cases). Should any isolated exceptions appear, they will be catalogued and shown to form a measure-zero set that does not affect the topological or measure-theoretic equivalence; otherwise the claim of completeness will be retained with the added verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The paper constructs substitution rules on FC words defined to act in parallel to the generating maps on positive reals for the permuted SB tree, then shows these induce a self-map on the space of infinite Sturmian sequences, and obtains a correspondence to geodesic and horocyclic motions. These steps are presented as explicit new constructions that enrich an existing correspondence, continuing contextually from [BI] but without the central claims reducing to fitted inputs, self-definitions, or unverified self-citations by construction. The parallelism and induced map are demonstrated through the rules themselves rather than tautologically assumed, and the geometric correspondence follows from the constructions without circular reduction to prior inputs. The paper remains self-contained in its original contributions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of Sturmian sequences, the Stern-Brocot tree, and Farey sequences together with the assumption that the new substitution rules preserve the parallel action with the real-line maps.

axioms (2)
  • domain assumption Sturmian sequences are closed under the induced substitution map
    Invoked when the rules are said to induce a map on the space of infinite Sturmian sequences
  • standard math The Stern-Brocot tree admits vertical and horizontal permutations generated by explicit maps on positive reals
    Background fact used to define the parallel action of the word substitutions

pith-pipeline@v0.9.0 · 5486 in / 1349 out tokens · 31352 ms · 2026-05-10T17:30:51.748984+00:00 · methodology

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Reference graph

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