Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On the partitions with Sturmian-like refinements

Pages: 3483 - 3501, Volume 35, Issue 8, August 2015      doi:10.3934/dcds.2015.35.3483

       Abstract        References        Full Text (421.3K)       Related Articles       

Michal Kupsa - Institute of Information Theory and Automation, The Academy of Sciences of the Czech Republic, Prague 8, CZ-18208, Czech Republic (email)
Štěpán Starosta - Faculty of Information Technology, Czech Technical University in Prague, Prague 6, CZ-16000, Czech Republic (email)

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.

Keywords:  Coding of rotation, Sturmian subshift, Toeplitz subshift, factor mapping, low-complexity system, sliding block-code, Sturmian partition, local rule.
Mathematics Subject Classification:  37B10, 68R15.

Received: April 2014;      Revised: December 2014;      Available Online: February 2015.