On the partitions with Sturmian-like refinements

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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

Abstract
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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 and 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.
[2]	P. Alessandri and V. Berthé, Three distance theorems and combinatorics on words, Enseign. Math. (2), 44 (1998), 103-132.
[3]	P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith., 87 (1999), 209-217.
[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.
[5]	P. Dartnell, F. Durand and A. Maass, Orbit equivalence and Kakutani equivalence with Sturmian subshifts, Studia Math., 142 (2000), 25-45.
[6]	G. Didier, Combinatoire des codages de rotations, Acta Arith., 85 (1998), 157-177.
[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.
[8]	P. N. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag Berlin Heidelberg, 2002. doi: 10.1007/b13861.
[9]	P. Kůrka, Topological and Symbolic Dynamics, Société Mathématique de France, Marseilles, 2003.
[10]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. doi: 10.1017/CBO9780511626302.
[11]	M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.
[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.

show all references

References:

[1]	P. Alessandri, Codages de Rotations et Basses Complexités, PhD thesis, Université d'Aix-Marseille II, 1996.
[2]	P. Alessandri and V. Berthé, Three distance theorems and combinatorics on words, Enseign. Math. (2), 44 (1998), 103-132.
[3]	P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith., 87 (1999), 209-217.
[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.
[5]	P. Dartnell, F. Durand and A. Maass, Orbit equivalence and Kakutani equivalence with Sturmian subshifts, Studia Math., 142 (2000), 25-45.
[6]	G. Didier, Combinatoire des codages de rotations, Acta Arith., 85 (1998), 157-177.
[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.
[8]	P. N. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag Berlin Heidelberg, 2002. doi: 10.1007/b13861.
[9]	P. Kůrka, Topological and Symbolic Dynamics, Société Mathématique de France, Marseilles, 2003.
[10]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. doi: 10.1017/CBO9780511626302.
[11]	M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.
[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.

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