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

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, 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.
Keywords: Sturmian subshift, Coding of rotation, local rule., low-complexity system, Sturmian partition, Toeplitz subshift, sliding block-code, factor mapping.
Mathematics Subject Classification: 37B10, 68R1.
Citation: Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483
References:
[1]

P. Alessandri, Codages de Rotations et Basses Complexités, PhD thesis, Université d'Aix-Marseille II, 1996. Google Scholar

[2]

P. Alessandri and V. Berthé, Three distance theorems and combinatorics on words, Enseign. Math. (2), 44 (1998), 103-132.  Google Scholar

[3]

P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith., 87 (1999), 209-217.  Google Scholar

[4]

J. Cassaigne and J. Karhumäki, Toeplitz words, generalized periodicity and periodically iterated morphisms, Eur. J. Comb., 18 (1997), 497-510. 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), 25-45.  Google Scholar

[6]

G. Didier, Combinatoire des codages de rotations, Acta Arith., 85 (1998), 157-177.  Google Scholar

[7]

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergod. Theor. Dyn. Syst., 20 (2000), 1061-1078. 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, Marseilles, 2003.  Google Scholar

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D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. doi: 10.1017/CBO9780511626302.  Google Scholar

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M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.  Google Scholar

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V. T. Sós, On the distribution mod 1 of the sequences $n\alpha$, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., 1 (1958), 127-134. Google Scholar

show all references

References:
[1]

P. Alessandri, Codages de Rotations et Basses Complexités, PhD thesis, Université d'Aix-Marseille II, 1996. Google Scholar

[2]

P. Alessandri and V. Berthé, Three distance theorems and combinatorics on words, Enseign. Math. (2), 44 (1998), 103-132.  Google Scholar

[3]

P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith., 87 (1999), 209-217.  Google Scholar

[4]

J. Cassaigne and J. Karhumäki, Toeplitz words, generalized periodicity and periodically iterated morphisms, Eur. J. Comb., 18 (1997), 497-510. 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), 25-45.  Google Scholar

[6]

G. Didier, Combinatoire des codages de rotations, Acta Arith., 85 (1998), 157-177.  Google Scholar

[7]

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergod. Theor. Dyn. Syst., 20 (2000), 1061-1078. 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, Marseilles, 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), 1-42. 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, Sect. Math., 1 (1958), 127-134. Google Scholar

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