[Paper Review] The structure of low complexity subshifts

Bastíán Espinoza|arXiv (Cornell University)|May 4, 2023

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper resolves the long-standing $σ$-adic conjecture by providing a complete $σ$-adic characterization for minimal subshifts of linear- and nonsuperlinear-complexity growth. It establishes that such subshifts admit an $σ$-adic sequence with bounded root word count and controlled length ratios, unifying access to $σ$-adic machinery and simplifying proofs of prior results like Cassaigne's Theorem.

ABSTRACT

An idea that became unavoidable to study zero entropy symbolic dynamics is that the dynamical properties of a system induce in it a combinatorial structure. An old problem addressing this intuition is finding a structure theorem for linear-growth complexity subshifts using the S-adic formalism. It is known as the S-adic conjecture and motivated several influential results in the theory. In this article, we completely solve the conjecture by providing an S-adic structure for this class. Our methods extend to nonsuperlinear-complexity subshifts. An important consequence of our main results is that these complexity classes gain access to the S-adic machinery. We show how this provides a unified framework and simplified proofs of several known results, including the pioneering 1996 Cassaigne's Theorem.

Motivation & Objective

• To resolve the $σ$-adic conjecture, which posits a structural characterization of minimal subshifts with linear-growth complexity using $σ$-adic sequences.
• To address the challenge of defining non-tautological conditions for such a characterization, avoiding trivial solutions that merely restate the complexity condition.
• To extend the $σ$-adic framework to nonsuperlinear-complexity subshifts, broadening its applicability beyond linear growth.
• To demonstrate that the new characterization enables unified, simplified proofs of known results, such as Cassaigne’s Theorem.
• To establish that minimal subshifts of nonsuperlinear-complexity growth have finite topological rank, leveraging the new $σ$-adic structure.

Proposed method

• Introduces a new $σ$-adic characterization using three key conditions: bounded root word count, controlled length ratios between images of letters, and bounded image lengths at each stage.
• Applies a desubstitution process via $σ$-adic sequences $(σ_n: \mathcal{A}_{n+1} \to \mathcal{A}_n^+)$ to represent subshifts with low complexity.
• Uses combinatorial bounds on word sets $\mathcal{V}_{n,w}$ and $\mathcal{W}_n$ to control complexity growth and ensure uniform bounds on the language of the subshift.
• Employs the concept of $\operatorname{\mathsf{root}}(w)$, the shortest primitive root of a word $w$, to constrain repetition and structure in the substitution process.
• Applies results from prior work (e.g., [Esp22, DDMP21]) to deduce finite topological rank from the new $σ$-adic structure.
• Leverages the divergence of $\langle\sigma_{[0,n)}\rangle$ to ensure that the root word sets grow in length, enabling application of rigidity theorems from the literature.

Experimental results

Research questions

• RQ1 Can a complete $σ$-adic characterization be given for minimal subshifts of linear-growth complexity, resolving the $σ$-adic conjecture?
• RQ2 What non-tautological conditions on $σ$-adic sequences ensure that the generated subshift has linear or nonsuperlinear complexity?
• RQ3 Does the new $σ$-adic structure unify and simplify existing proofs of classical results in zero entropy symbolic dynamics?
• RQ4 Can the $σ$-adic framework be extended to nonsuperlinear-complexity subshifts, and what structural constraints emerge?
• RQ5 Does every minimal subshift of nonsuperlinear-complexity growth have finite topological rank, and can this be proven via the new $σ$-adic characterization?

Key findings

• The paper provides a complete solution to the $σ$-adic conjecture: a minimal subshift has linear-growth complexity if and only if it is generated by a $σ$-adic sequence satisfying conditions $(\mathcal{P}_1)$, $(\mathcal{P}_2)$, and $(\mathcal{P}_3)$.
• For nonsuperlinear-complexity subshifts, a similar characterization holds with only $(\mathcal{P}_1)$ and $(\mathcal{P}_2)$, showing broader applicability of the framework.
• The language complexity $p_S(k)$ of the base set $S$ is uniformly bounded, with $p_S(k) \leq (\log_2 d + 7) \cdot 2^{12} d^4$, ensuring structural finiteness.
• The new $σ$-adic structure implies that all minimal subshifts of nonsuperlinear-complexity growth have finite topological rank, confirming a result from [DDMP21] with a new proof.
• The framework provides a unified approach that simplifies proofs of known results, including Cassaigne’s Theorem, by embedding them into a coherent $σ$-adic setting.
• The proof relies on bounding word sets $\mathcal{V}_{n,w}$ and using length comparisons between $\sigma_{[0,n)}(a)$ and $\sigma_{[0,n)}(b)$ to control complexity growth.