[Paper Review] Measure-Theoretically Mixing Subshifts of Minimal Word Complexity

Darren Creutz|arXiv (Cornell University)|Jun 20, 2022

semigroups and automata theory4 citations

TL;DR

This paper resolves a longstanding question in symbolic dynamics by establishing that superlinear word complexity is the precise threshold for measure-theoretic strong mixing in subshifts. It constructs subshifts with word complexity arbitrarily close to linear that admit strongly mixing measures, while proving that non-superlinear complexity implies partial rigidity and thus non-mixing.

ABSTRACT

We resolve a long-standing open question on the relationship between measure-theoretic dynamical complexity and symbolic complexity by establishing the exact word complexity at which measure-theoretic strong mixing manifests: For every superlinear $f : \mathbb{N} o \mathbb{N}$, i.e. $f(q)/q o \infty$, there exists a subshift admitting a (strongly) mixing of all orders probability measure with word complexity $p$ such that $p(q)/f(q) o 0$. For a subshift with word complexity $p$ which is non-superlinear, i.e. $\liminf p(q)/q < \infty$, every ergodic probability measure is partially rigid.

Motivation & Objective

• To resolve the open question of the minimal word complexity at which measure-theoretic strong mixing can occur in subshifts.
• To demonstrate that subshifts with word complexity growing faster than any linear function (but arbitrarily slowly superlinear) can still admit strongly mixing measures.
• To prove that subshifts with non-superlinear word complexity (i.e., liminf p(q)/q < ∞) cannot be strongly mixing, as all ergodic measures are partially rigid.
• To clarify the sharp dichotomy between structural complexity (S-adic, partial rigidity) and dynamical complexity (strong mixing) at the superlinear threshold.

Proposed method

• Constructs a new class of rank-one transformations called 'quasi-staircase' transformations to achieve mixing with subquadratic but superlinear word complexity.
• Uses a recursive construction based on interval exchange and tower-building techniques to control word complexity while ensuring mixing properties.
• Applies measure-theoretic estimates involving cylinder sets and return times to verify strong mixing of all orders.
• Employs a recursive argument over levels of the tower to ensure positive measure intersections over long return times, confirming partial rigidity in the non-superlinear case.
• Uses the Cassaigne formula for word complexity: p(q) = p(m) + ∑_{ℓ=m}^{q−1} |{w ∈ L^{RS} : ||w|| = ℓ}| to analyze complexity growth.
• Applies the Mean Ergodic Theorem and measure-theoretic isomorphism arguments to relate structural properties of the subshift to its dynamical behavior.

Experimental results

Research questions

• RQ1 What is the minimal word complexity function p(q) for which a subshift can admit a strongly mixing invariant measure?
• RQ2 Can strongly mixing measures exist in subshifts with word complexity arbitrarily close to linear?
• RQ3 Is there a sharp threshold in word complexity that separates partially rigid systems from strongly mixing ones?
• RQ4 How does non-superlinear word complexity relate to the existence of ergodic measures with partial rigidity?
• RQ5 To what extent do S-adic structures constrain the dynamical complexity of subshifts?

Key findings

• For every superlinear function f: ℕ → ℕ with f(q)/q → ∞, there exists a subshift with word complexity p(q) such that p(q)/f(q) → 0 and admitting a strongly mixing measure of all orders.
• The constructed examples are rank-one transformations, hence mixing of all orders, and achieve word complexity arbitrarily close to linear growth.
• If a subshift has non-superlinear word complexity (i.e., liminf p(q)/q < ∞), then every ergodic measure is partially rigid and thus not strongly mixing.
• This partial rigidity implies that such subshifts are conjugate to S-adic shifts, which are known to be highly structured.
• The threshold at superlinear complexity separates systems with at most countably many ergodic measures (non-superlinear) from those with uncountably many (superlinear), aligning with results by Cyr and Kra.
• The paper establishes that superlinear word complexity is the precise dividing line between structural (S-adic, partially rigid) and complex (strongly mixing) dynamical behavior.