On the partitions with Sturmian-like refinements

Michal Kupsa; Štěpán Starosta

doi:10.3934/dcds.2015.35.3483

Article Contents

2015, Volume 35, Issue 8: 3483-3501. Doi: 10.3934/dcds.2015.35.3483

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On the partitions with Sturmian-like refinements

Michal Kupsa^1, and
Štěpán Starosta^2,

1.
Institute of Information Theory and Automation, The Academy of Sciences of the Czech Republic, Prague 8, CZ-18208
2.
Faculty of Information Technology, Czech Technical University in Prague, Prague 6, CZ-16000

Received: March 31, 2014

Revised: November 30, 2014

Published: May 2017

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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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Mathematics Subject Classification: 37B10, 68R15.

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