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arXiv · 1201.5757

Lower bounds for symbolic complexity of iceberg dynamical systems

A. A. Prikhod'ko
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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 $l_n = |w_n|+1$.

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A. A. Prikhod'ko. 2012-01-27. Lower bounds for symbolic complexity of iceberg dynamical systems. https://arxiv.org/abs/1201.5757

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