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arXiv · 1303.3387

On the partitions with Sturmian-like refinements

Michal Kupsa·Štepán Starosta
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Abstract

In the dynamics of a rotation of the unit circle by an irrational angle $\alpha\in(0,1)$, we study the evolution of partitions whose atoms are finite unions of left-closed right-open intervals with endpoints lying on the past trajectory of the point $0$. Unlike the standard framework, we focus on partitions whose atoms are disconnected sets. We show that the refinements of these partitions eventually coincide with the refinements of a preimage of the Sturmian partition, which consists of two intervals $[0,1-\alpha)$ and $[1-\alpha,1)$. In particular, the refinements of the partitions eventually consist of connected sets, i.e., intervals. We reformulate this result in terms of Sturmian subshifts: we show that for every non-trivial factor mapping from a one-sided Sturmian subshift, satisfying a mild technical assumption, the sliding block code of sufficiently large length induced by the mapping is injective.

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Michal Kupsa, Štepán Starosta. 2013-03-14. On the partitions with Sturmian-like refinements. https://doi.org/10.3934/dcds.2015.35.3483

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