[Paper Review] Computations for symbolic substitutions

D. M. Rust, Scott Balchin|arXiv (Cornell University)|May 30, 2017

semigroups and automata theory28 references3 citations

TL;DR

This paper presents Grout, a user-friendly computational tool for analyzing symbolic substitutions in symbolic dynamics, focusing on computing topological invariants like cohomology and testing the strong coincidence conjecture. The authors implement a deterministic algorithm to check eigenvalue modulus relative to the unit circle and perform a large-scale search across over 300,000 three-letter and 200,000 four-letter substitutions, finding no counterexamples to the strong coincidence conjecture.

ABSTRACT

We provide a survey of results from symbolic dynamics and algebraic topology relating to Grout, a new user-friendly program developed to calculate combinatorial properties and topological invariants of a large class of symbolic substitutions. We study their subshifts (and related spaces) with an emphasis on examples of computations. We implement a check to verify that no counterexample exists to the so-called "strong coincidence conjecture" for a large number of substitutions on three and four letters.

Motivation & Objective

• To develop a user-friendly computational tool, Grout, for analyzing symbolic substitutions and their topological invariants.
• To implement a robust, deterministic algorithm for checking whether eigenvalues of substitution matrices lie on, inside, or outside the unit circle.
• To perform a large-scale computational search for counterexamples to the strong coincidence conjecture among irreducible Pisot substitutions on three and four letters.
• To support theoretical research in symbolic dynamics and tiling theory by enabling efficient verification of conjectures and automation of diagram generation.
• To provide a scalable, accessible platform for researchers to explore combinatorial and topological properties of substitution systems.

Proposed method

• Implement a deterministic numerical check for eigenvalue modulus relative to the unit circle, overcoming limitations of standard floating-point eigenvalue solvers by using a threshold-based convergence method.
• Use lexicographical ordering of substitution rules (modulo permutations and reversals) to systematically enumerate substitutions over three and four-letter alphabets.
• Apply standard linear algebra techniques to compute eigenvalues and check irreducibility over ℤ for substitution matrices.
• Utilize the Anderson-Putnam complex construction to compute the Čech cohomology of tiling spaces associated with substitutions.
• Integrate a recognizability check for primitive substitutions using a fully deterministic algorithm suitable for both hand calculation and programmatic use.
• Automate the generation of TikZ code for topological complexes and export results to LaTeX for typesetting, enhancing reproducibility and visualization.

Experimental results

Research questions

• RQ1 Can a computational tool be developed that efficiently computes cohomology and other invariants for symbolic substitutions with high usability?
• RQ2 Does the strong coincidence conjecture hold for all irreducible Pisot substitutions on three and four letters?
• RQ3 Can numerical eigenvalue computation be made reliable for boundary cases where eigenvalues lie exactly on the unit circle?
• RQ4 Is there a finite bound on the number of iterations required for a strong coincidence to emerge in any substitution system?
• RQ5 Can patterns in the number of iterations to coincidence formation suggest the existence of infinite families of substitutions requiring increasingly many steps?

Key findings

• No counterexample to the strong coincidence conjecture was found in the first 321,425,442 three-letter substitutions, of which 12,573,955 were irreducible Pisot.
• No counterexample to the strong coincidence conjecture was found in the first 200,399 four-letter substitutions, of which 15,001 were irreducible Pisot.
• The maximum number of iterations required to achieve a strong coincidence across all tested substitutions was 11, with only two substitutions requiring 10 iterations.
• The distribution of iterations shows a sharp decline: 1,921,144 substitutions achieved coincidence in 1 iteration, while only 3 substitutions required 9 iterations.
• The two substitutions requiring 10 iterations were explicitly listed: 0↦2011, 1↦02, 2↦0 and 0↦212101, 1↦0, 2↦1.
• No clear pattern emerged from the 'record-breaking' substitutions that would allow construction of an infinite family requiring arbitrarily many iterations.