American Institute of Mathematical Sciences

Advanced
October  2015, 35(10): 4743-4764. doi: 10.3934/dcds.2015.35.4743

Transitive sofic spacing shifts

John Banks 1, , Piotr Oprocha 2, and  Brett Stanley 3,

1. 

Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Australia
2. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków
3. 

Department of Mathematics and Statistics, La Trobe University, Bundoora 3086, Australia

Received  May 2014 Revised  February 2015 Published  April 2015

Spacing shifts were introduced by Lau and Zame in the 1970's to provide accessible examples of maps that are weakly mixing but not mixing. In previous papers by the authors and others, it has been observed that the problem of describing when spacing shifts are topologically transitive appears to be quite difficult in general. In the present paper, we give a characterization of sofic spacing shifts and begin to investigate which sofic spacing shifts are topologically transitive. We show that the canonical graph presentation of such a shift has a rather simple form, for which we introduce the terminology hereditary bunched cycle and discuss the apparently difficult problem of determining which hereditary bunched cycles actually present spacing shifts.
Keywords: Symbolic dynamics, sofic shifts, topological transitivity, spacing shifts..
Mathematics Subject Classification: Primary: 37B10; Secondary: 37B20, 05C38, 68Q4.
Citation: John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743
References:
[1]

D. Ahmadi and M. Dabbaghian, Characterization of spacing shifts with positive topological entropy,, Acta Math. Univ. Comenian. (N.S.), 81 (2012), 221. Google Scholar

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances,, North-Holland Mathematical Library, (1994). Google Scholar

[3]

J. Banks, Regular Periodic Decompositions for Topologically Transitive maps,, Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 505. doi: 10.1017/S0143385797069885. Google Scholar

[4]

J. Banks, T. T. D. Nguyen, P. Oprocha, B. Stanley and B. Trotta, Dynamics of spacing shifts,, Discrete Contin. Dyn. Syst. 33 (2013), 33 (2013), 4207. doi: 10.3934/dcds.2013.33.4207. Google Scholar

[5]

D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition,, John Wiley & Sons, (2004). Google Scholar

[6]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981). Google Scholar

[7]

S. Ginsburg, Algebraic and automata-theoretic properties of formal languages,, North-Holland/American Elsevier, (1975). Google Scholar

[8]

M. Harrison, Introduction to Formal Language Theory,, Addison-Wesley, (1978). Google Scholar

[9]

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts,, Discrete Contin. Dyn. Syst, 33 (2013), 2451. doi: 10.3934/dcds.2013.33.2451. Google Scholar

[10]

K. Lau and A. Zame, On weak mixing of cascades,, Math. Systems Theory, 6 (1973), 307. doi: 10.1007/BF01740722. Google Scholar

[11]

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

[12]

M. Lothaire, Algebraic Combinatorics on Words,, Cambridge University Press, (1997). doi: 10.1017/CBO9780511566097. Google Scholar

show all references

References:
[1]

D. Ahmadi and M. Dabbaghian, Characterization of spacing shifts with positive topological entropy,, Acta Math. Univ. Comenian. (N.S.), 81 (2012), 221. Google Scholar

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances,, North-Holland Mathematical Library, (1994). Google Scholar

[3]

J. Banks, Regular Periodic Decompositions for Topologically Transitive maps,, Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 505. doi: 10.1017/S0143385797069885. Google Scholar

[4]

J. Banks, T. T. D. Nguyen, P. Oprocha, B. Stanley and B. Trotta, Dynamics of spacing shifts,, Discrete Contin. Dyn. Syst. 33 (2013), 33 (2013), 4207. doi: 10.3934/dcds.2013.33.4207. Google Scholar

[5]

D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition,, John Wiley & Sons, (2004). Google Scholar

[6]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981). Google Scholar

[7]

S. Ginsburg, Algebraic and automata-theoretic properties of formal languages,, North-Holland/American Elsevier, (1975). Google Scholar

[8]

M. Harrison, Introduction to Formal Language Theory,, Addison-Wesley, (1978). Google Scholar

[9]

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts,, Discrete Contin. Dyn. Syst, 33 (2013), 2451. doi: 10.3934/dcds.2013.33.2451. Google Scholar

[10]

K. Lau and A. Zame, On weak mixing of cascades,, Math. Systems Theory, 6 (1973), 307. doi: 10.1007/BF01740722. Google Scholar

[11]

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

[12]

M. Lothaire, Algebraic Combinatorics on Words,, Cambridge University Press, (1997). doi: 10.1017/CBO9780511566097. Google Scholar

[1]

John Banks, Thi T. D. Nguyen, Piotr Oprocha, Brett Stanley, Belinda Trotta. Dynamics of spacing shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4207-4232. doi: 10.3934/dcds.2013.33.4207
[2]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[3]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[4]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459
[5]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[6]

Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001
[7]

Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
[8]

John Banks, Brett Stanley. A note on equivalent definitions of topological transitivity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1293-1296. doi: 10.3934/dcds.2013.33.1293
[9]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[10]

Bing Li, Tuomas Sahlsten, Tony Samuel. Intermediate $\beta$-shifts of finite type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 323-344. doi: 10.3934/dcds.2016.36.323
[11]

Philipp Gohlke, Dan Rust, Timo Spindeler. Shifts of finite type and random substitutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5085-5103. doi: 10.3934/dcds.2019206
[12]

Filomena Garcia, Joana Resende. Conformity-based behavior and the dynamics of price competition: A new rationale for fashion shifts. Journal of Dynamics & Games, 2016, 3 (2) : 153-167. doi: 10.3934/jdg.2016008
[13]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003
[14]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131
[15]

Kengo Matsumoto. On the Markov-Dyck shifts of vertex type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 403-422. doi: 10.3934/dcds.2016.36.403
[16]

Kevin McGoff, Ronnie Pavlov. Random $\mathbb{Z}^d$-shifts of finite type. Journal of Modern Dynamics, 2016, 10: 287-330. doi: 10.3934/jmd.2016.10.287
[17]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[18]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[19]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[20]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences