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August  2015, 35(8): 3483-3501. doi: 10.3934/dcds.2015.35.3483

On the partitions with Sturmian-like refinements

 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, Czech Republic

Received  April 2014 Revised  December 2014 Published  February 2015

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.
Citation: Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483
References:
 [1] P. Alessandri, Codages de Rotations et Basses Complexités,, PhD thesis, (1996).   Google Scholar [2] P. Alessandri and V. Berthé, Three distance theorems and combinatorics on words,, Enseign. Math. (2), 44 (1998), 103.   Google Scholar [3] P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations,, Acta Arith., 87 (1999), 209.   Google Scholar [4] J. Cassaigne and J. Karhumäki, Toeplitz words, generalized periodicity and periodically iterated morphisms,, Eur. J. Comb., 18 (1997), 497.  doi: 10.1006/eujc.1996.0110.  Google Scholar [5] P. Dartnell, F. Durand and A. Maass, Orbit equivalence and Kakutani equivalence with Sturmian subshifts,, Studia Math., 142 (2000), 25.   Google Scholar [6] G. Didier, Combinatoire des codages de rotations,, Acta Arith., 85 (1998), 157.   Google Scholar [7] F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors,, Ergod. Theor. Dyn. Syst., 20 (2000), 1061.  doi: 10.1017/S0143385700000584.  Google Scholar [8] P. N. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics,, Springer-Verlag Berlin Heidelberg, (2002).  doi: 10.1007/b13861.  Google Scholar [9] P. Kůrka, Topological and Symbolic Dynamics,, Société Mathématique de France, (2003).   Google Scholar [10] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511626302.  Google Scholar [11] M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1.  doi: 10.2307/2371431.  Google Scholar [12] V. T. Sós, On the distribution mod 1 of the sequences $n\alpha$,, Ann. Univ. Sci. Budap. Rolando Eötvös, 1 (1958), 127.   Google Scholar

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References:
 [1] P. Alessandri, Codages de Rotations et Basses Complexités,, PhD thesis, (1996).   Google Scholar [2] P. Alessandri and V. Berthé, Three distance theorems and combinatorics on words,, Enseign. Math. (2), 44 (1998), 103.   Google Scholar [3] P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations,, Acta Arith., 87 (1999), 209.   Google Scholar [4] J. Cassaigne and J. Karhumäki, Toeplitz words, generalized periodicity and periodically iterated morphisms,, Eur. J. Comb., 18 (1997), 497.  doi: 10.1006/eujc.1996.0110.  Google Scholar [5] P. Dartnell, F. Durand and A. Maass, Orbit equivalence and Kakutani equivalence with Sturmian subshifts,, Studia Math., 142 (2000), 25.   Google Scholar [6] G. Didier, Combinatoire des codages de rotations,, Acta Arith., 85 (1998), 157.   Google Scholar [7] F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors,, Ergod. Theor. Dyn. Syst., 20 (2000), 1061.  doi: 10.1017/S0143385700000584.  Google Scholar [8] P. N. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics,, Springer-Verlag Berlin Heidelberg, (2002).  doi: 10.1007/b13861.  Google Scholar [9] P. Kůrka, Topological and Symbolic Dynamics,, Société Mathématique de France, (2003).   Google Scholar [10] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511626302.  Google Scholar [11] M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1.  doi: 10.2307/2371431.  Google Scholar [12] V. T. Sós, On the distribution mod 1 of the sequences $n\alpha$,, Ann. Univ. Sci. Budap. Rolando Eötvös, 1 (1958), 127.   Google Scholar
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