[Paper Review] Symbolic coding for linear trajectories in the regular octagon

John Smillie, Corinna Ulcigrai|ArXiv.org|May 6, 2009

Mathematical Dynamics and Fractals16 references10 citations

TL;DR

This paper presents a symbolic coding system for linear trajectories in the regular octagon using cutting sequences—bi-infinite words representing the order in which trajectory paths cross labeled sides. It introduces a novel continued fraction algorithm tied to the Veech group's renormalization dynamics, which characterizes cutting sequences via substitution operations and establishes a complete, explicit correspondence between trajectory slopes and their symbolic sequences, generalizing Sturmian sequences to higher-order polygons.

ABSTRACT

We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well studied Sturmian sequences which can be analyzed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our continued fraction expansion of the slope gives an explicit sequence of substitution operations which generate the cutting sequences of trajectories with that slope. Our algorithm can be understood in terms of renormalization of the octagon translation surface by elements of the Veech group.

Motivation & Objective

• To provide a complete and explicit characterization of symbolic cutting sequences arising from linear trajectories in the regular octagon.
• To extend the theory of Sturmian sequences—used in the square case—to the regular octagon and, more generally, to regular 2n-gons.
• To establish a correspondence between the slope of a trajectory and its cutting sequence using a new continued fraction algorithm rooted in renormalization and Teichmüller dynamics.
• To generalize the symbolic coding framework to all regular polygons with an even number of sides, showing that their cutting sequences are characterized by infinite coherence and generation operators.

Proposed method

• Introduce a symbolic coding via cutting sequences, where each trajectory is mapped to a bi-infinite word over the alphabet {A, B, C, D} based on the sequence of side pairs it crosses.
• Develop a continued fraction algorithm tailored to the octagon’s geometry, analogous to the classical continued fraction approach for the square, but adapted to the Veech group’s action on the translation surface.
• Use renormalization via the Veech group to define a dynamical system that relates the slope of a trajectory to its cutting sequence through iterative substitution operations.
• Define generation operators 𝔤ⁱⱼ that act on words in the alphabet 𝒜ₛ, enabling recursive construction of all valid cutting sequences through a hierarchical, coherent structure.
• Represent the affine polygon O′₂ₙ as a staircase of rectangles to visualize and analyze sandwiching conditions, linking geometric angles to symbolic letter sequences.
• Establish coherence conditions (C1)′ and (C2)′ that characterize admissible finite subwords and ensure closure of the set of cutting sequences.

Experimental results

Research questions

• RQ1 Can we give a complete and explicit characterization of the infinite words that arise as cutting sequences of linear trajectories in the regular octagon?
• RQ2 Does a given cutting sequence uniquely determine the direction (slope) of the corresponding trajectory?
• RQ3 Given a finite subword of a cutting sequence, can we identify a sector of possible directions, and how can this be computed algorithmically?
• RQ4 How can the symbolic coding of trajectories in the octagon be generalized to all regular 2n-gons with even sides?
• RQ5 What is the role of the Veech group and Teichmüller geodesic flow in generating and classifying these cutting sequences?

Key findings

• The set of all cutting sequences for the regular octagon is completely characterized as the closure of the set of infinitely coherent words under the generation operators 𝔤ⁱⱼ.
• The paper establishes a one-to-one correspondence between the slope of a linear trajectory and its cutting sequence via a novel continued fraction algorithm derived from the Veech group’s action.
• The cutting sequences of trajectories in the regular octagon are generated by a sequence of substitution operations defined by the generation operators, generalizing the Sturmian case.
• For regular 2n-gons, the closure of the set of cutting sequences coincides with the set of infinitely coherent words, and is described by the nested intersection of iterated generation operator applications.
• The staircase representation of the affine polygon O′₂ₙ allows geometric interpretation of sandwiching conditions, linking side labels to the action of permutations π→ and π↑.
• The method provides a complete symbolic description of a measure-zero family of interval exchange transformations via a renormalization scheme that captures their dynamics through coherent symbolic sequences.