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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, 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), 221-226.  Google Scholar

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52, North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar

[3]

J. Banks, Regular Periodic Decompositions for Topologically Transitive maps, Ergodic Theory Dynam. Systems 17 (1997), 505-529. 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), 4207-4232. doi: 10.3934/dcds.2013.33.4207.  Google Scholar

[5]

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

[6]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 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), 2451-2467. 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), 307-311. 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), 221-226.  Google Scholar

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52, North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar

[3]

J. Banks, Regular Periodic Decompositions for Topologically Transitive maps, Ergodic Theory Dynam. Systems 17 (1997), 505-529. 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), 4207-4232. doi: 10.3934/dcds.2013.33.4207.  Google Scholar

[5]

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

[6]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 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), 2451-2467. 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), 307-311. 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

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