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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