Christian Steineder: Complexity of Hartman sequences

This is a joint work with Reinhard Winkler.

Let $T: x \mapsto x+g$ be an ergodic translation on the compact group $C$ and $M \subseteq C$ a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence $\mbox{$\bf {a}$}$$:$   $\mbox{$\bf {Z}$}$$\mapsto \{0,1\}$ defined by $\mbox{$\bf {a}$}$$(k) =1$ if $T^k(0_C) \in M$ and $\mbox{$\bf {a}$}$$(k) =0$ otherwise, is called a Hartman sequence. In this talk we will study the growth rate of $P_{\mbox{$\bf {a}$}}(n)$, where $P_{\mbox{$\bf {a}$}}(n)$ denotes the number of binary words of length $n \in$   $\mbox{$\bf {N}$}$ occurring in $\mbox{$\bf {a}$}$. For the case that $T$ is an ergodic translation $x \mapsto x + \alpha$ $(\alpha =(\alpha_1,\ldots,\alpha_s))$ on the $s$-dimensional torus we present a geometric method to calculate the asymptotic behaviour of $P_{\mbox{$\bf {a}$}}(n)$.

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