[Paper Review] Lower bounds for symbolic complexity of iceberg dynamical systems

A. A. Prikhod’ko|arXiv (Cornell University)|Jan 27, 2012

semigroups and automata theory14 references3 citations

TL;DR

This paper establishes a lower bound of order $ l^{3- u} $ for the symbolic complexity of iceberg dynamical systems—generalized rank-one systems constructed via random cyclic rotations of words—demonstrating they exceed the quadratic complexity typical of classical rank-one systems. The result implies that iceberg systems form a strictly larger class than rank-one maps, with implications for spectral invariants and isomorphism problems in zero entropy dynamics.

ABSTRACT

The symbolic complexity of an infinite word $W$ is the function $p_W(l)$ counting the number of different subwords in $W$ of length $l$. In this paper our main purpose is to study the complexity for a class of topological dynamical systems, called iceberg systems, given by the following symbolic procedure. Starting from a given finite word $w_1$ we construct a sequence of words $w_{n+1} = w_n ρ_{a_n(1)}(w_n)...ρ_{a_n(q_n-1)}(w_n)$, where $ρ_a(u)$ is the cyclic rotations of the word $u$ by $a$ positions, and consider an infinite word $W$ extending each $w_n$ to the right. It is shown that for iceberg systems given by the randomized parameters $a_n(j)$ the complexity function almost surely satisfies the estimate $p_W(l) > l^{3-ε}$ for any $ε> 0$ and $l \ge l_0(ε)$, and at the same time it is observed that this estimate represents up to a small correction the optimal lower bound for the complexity function, namely, $p_{w_{n+1}}(l_n) \le l_n^3$ along the subsequence $l_n = |w_n|+1$.

Motivation & Objective

• To investigate the symbolic complexity of a generalized class of dynamical systems called iceberg systems, extending classical rank-one constructions.
• To determine whether iceberg systems exhibit higher complexity than classical rank-one systems, which are known to have $ p(l) \lesssim l^2 $.
• To explore the role of symbolic complexity as an invariant to distinguish between isomorphic but non-isomorphic systems, particularly in zero entropy dynamics.
• To compare the complexity of iceberg systems with that of the Pascal adic transformation, which has cubic complexity $ \sim l^3/6 $.

Proposed method

• Constructs iceberg systems via iterative word extension: $ \boldsymbol{w}_{n+1} = \boldsymbol{w}_n \rho_{\alpha_n(1)}(\boldsymbol{w}_n) \cdots \rho_{\alpha_n(q_n-1)}(\boldsymbol{w}_n) $, where $ \rho_\alpha $ denotes cyclic rotation.
• Analyzes the symbolic complexity function $ p_{\boldsymbol{w}_\infty}(l) $, counting distinct subwords of length $ l $ in the infinite limit word $ \boldsymbol{w}_\infty $.
• Uses randomized rotation parameters $ \alpha_n(j) $ to model typical behavior and derive probabilistic lower bounds on complexity.
• Establishes a subsequence $ l_n = |\boldsymbol{w}_n| + 1 $ where $ p_{\boldsymbol{w}_{n+1}}(l_n) \leq l_n^3 $, showing the cubic bound is tight up to small corrections.
• Applies scaling approximation theory to show iceberg maps have scaling rank one with $ \boldsymbol{\lambda}(h) = 1/2 $, contrasting with $ \boldsymbol{\lambda}(h) = 1 $ for rank-one maps.
• Compares results with known complexity of the Pascal adic transformation, which achieves $ p(l) \sim l^3/6 $.

Experimental results

Research questions

• RQ1 What is the typical symbolic complexity of iceberg systems constructed with random cyclic rotations?
• RQ2 Can iceberg systems be distinguished from classical rank-one systems using symbolic complexity as an invariant?
• RQ3 Is the cubic growth rate $ l^3 $ optimal for iceberg systems, or can tighter bounds be established?
• RQ4 How does the scaling function $ \boldsymbol{\lambda}(h) $ of iceberg systems compare to that of the Pascal adic transformation?
• RQ5 Is the asymptotic complexity $ \sim l^3/6 $ of the Pascal adic map optimal, and does it reflect a universal scaling behavior?

Key findings

• For iceberg systems with randomized rotation parameters, the symbolic complexity satisfies $ p_{\boldsymbol{w}_\infty}(l) \gtrsim l^{3-\varepsilon} $ for any $ \varepsilon > 0 $, indicating near-cubic growth.
• The upper bound $ p_{\boldsymbol{w}_{n+1}}(l_n) \leq l_n^3 $ along the subsequence $ l_n = |\boldsymbol{w}_n| + 1 $ shows that the cubic bound is nearly optimal.
• Iceberg systems are strictly more complex than classical rank-one systems, since the latter satisfy $ p(l_n) \lesssim \frac{1}{2} l_n^2 $, contradicting the $ l^{3-\varepsilon} $ lower bound.
• Iceberg maps are of scaling rank one with $ \boldsymbol{\lambda}(h) = 1/2 $, reflecting their construction via two-part interval exchanges (cyclic rotations).
• The symbolic complexity of the Pascal adic transformation is exactly $ p(l) \sim \frac{1}{6} l^3 $, making it a natural benchmark for comparing iceberg systems.
• The results suggest that symbolic complexity can distinguish non-isomorphic systems even when spectral properties are similar, implying rank is not a spectral invariant.