Words and numbers: a dynamical systems perspective

TL;DR

This paper explores the deep connections between combinatorics of words, rational number orderings, and dynamical systems, introducing new substitution rules and mappings related to Farey-Christoffel words and geometric structures.

Contribution

It introduces new substitution rules acting on Farey-Christoffel words and establishes a comprehensive correspondence between rational orderings, dynamical maps, and geometric motions in the upper half-plane.

Findings

Constructed substitution rules for Farey-Christoffel words.

Mapped the space of Sturmian sequences into itself.

Linked SB tree motions with geodesic and horocyclic motions in hyperbolic geometry.

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.