Lower bounds for symbolic complexity of iceberg dynamical systems

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 $w_{n+1} = w_n \rho_{a_n(1)}(w_n)...\rho_{a_n(q_n-1)}(w_n)$, where $\rho_a(u)$ 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 $p_W(l) > l^{3-\epsilon}$ for any $\epsilon > 0$ and $l \ge l_0(\epsilon)$, 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 $l_n = |w_n|+1$.