American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source
  • DCDS Home
  • This Issue
  • Next Article
    Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains
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 - 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

show all references

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

[1]

Jeanette Olli. Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4173-4186. doi: 10.3934/dcds.2013.33.4173
[2]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[3]

Bin Li, Hai Huyen Dam, Antonio Cantoni. A low-complexity zero-forcing Beamformer design for multiuser MIMO systems via a dual gradient method. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012
[4]

Silvère Gangloff, Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1975-2000. doi: 10.3934/dcds.2019083
[5]

Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
[6]

David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287
[7]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
[8]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315
[9]

Gokhan Calis, O. Ozan Koyluoglu. Architecture-aware coding for distributed storage: Repairable block failure resilient codes. Advances in Mathematics of Communications, 2018, 12 (3) : 465-503. doi: 10.3934/amc.2018028
[10]

Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[11]

Afaf Bouharguane, Pascal Azerad, Frédéric Bouchette, Fabien Marche, Bijan Mohammadi. Low complexity shape optimization & a posteriori high fidelity validation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 759-772. doi: 10.3934/dcdsb.2010.13.759
[12]

Wei Wan, Haiyang Huang, Jun Liu. Local block operators and TV regularization based image inpainting. Inverse Problems & Imaging, 2018, 12 (6) : 1389-1410. doi: 10.3934/ipi.2018058
[13]

Ayla Sayli, Ayse Oncu Sarihan. Statistical query-based rule derivation system by backward elimination algorithm. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1341-1356. doi: 10.3934/dcdss.2015.8.1341
[14]

Natalie Priebe Frank, Lorenzo Sadun. Topology of some tiling spaces without finite local complexity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 847-865. doi: 10.3934/dcds.2009.23.847
[15]

Jeong-Yup Lee, Boris Solomyak. On substitution tilings and Delone sets without finite local complexity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3149-3177. doi: 10.3934/dcds.2019130
[16]

Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545
[17]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[18]

Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz, Huaiping Zhu. How seasonal forcing influences the complexity of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 785-807. doi: 10.3934/dcdsb.2018043
[19]

Huijuan Li, Robert Baier, Lars Grüne, Sigurdur F. Hafstein, Fabian R. Wirth. Computation of local ISS Lyapunov functions with low gains via linear programming. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2477-2495. doi: 10.3934/dcdsb.2015.20.2477
[20]

Ying Zhang. Wave breaking and global existence for the periodic rotation-Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2243-2257. doi: 10.3934/dcds.2017097

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences