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January  2019, 39(1): 1-18. doi: 10.3934/dcds.2019001

Markov-Dyck shifts, neutral periodic points and topological conjugacy

Wolfgang Krieger 1, and  Kengo Matsumoto 2,

1. 

Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
2. 

Department of Mathematics, Joetsu University of Education, Joetsu 943 - 8512, Japan

Received  May 2017 Revised  June 2018 Published  October 2018

We study the neutral periodic points of Markov-Dyck shifts of finite strongly connected directed graphs. Under certain hypothesis on the structure of the graphs $G$ we show, that the topological conjugacy of their Markov-Dyck shifts implies the isomorphism of the graphs.

Keywords: Symbolic dynamics, Markov-Dyck shift, neutral periodic point, topological conjugacy, graph reconstruction.
Mathematics Subject Classification: 37B10.
Citation: Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001
References:
[1]

A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shifts, Ergod. Th. & Dynam. Sys., 36 (2016), 470-513.  doi: 10.1017/etds.2014.74.  Google Scholar

[2]

G. Gordon and E. McMahon, A greedoid polynomial which distinguishes rooted arborescences, Proc. AMS., 107 (1989), 287-298.  doi: 10.1090/S0002-9939-1989-0967486-0.  Google Scholar

[3]

T. Hamachi and K. Inoue, Embeddings of shifts of finite type into the Dyck shift, Monatsh. Math., 145 (2005), 107-129.  doi: 10.1007/s00605-004-0297-5.  Google Scholar

[4]

T. Hamachi, K. Inoue and W. Krieger, Subsystems of finite type and semigroup invariants of subshifts, J. reine angew. Math., 632 (2009), 37-61.  doi: 10.1515/CRELLE.2009.049.  Google Scholar

[5]

T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups, arXiv: 1303.4158 [math.DS] Google Scholar

[6]

B. P. Kitchens, Symbolic Dynamics, Springer, Berlin, Heidelberg, New York, 1998.  Google Scholar

[7]

W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104.  doi: 10.1007/BF01762180.  Google Scholar

[8]

W. Krieger, On a syntactically defined invariant of symbolic dynamics, Ergod. Th. & Dynam.Sys, 10 (2000), 501-506.  doi: 10.1017/S0143385700000249.  Google Scholar

[9]

W. Krieger, On subshifts and semigroups, Bull. London Math. Soc., 38 (2006), 617-624.  doi: 10.1112/S0024609306018625.  Google Scholar

[10]

W. Krieger, On flow equivalence of R-graph shifts, Münster J. Math., 8 (2015), 229-239.   Google Scholar

[11]

W. Krieger, On subshift presentations, Ergod. Th. & Dynam. Sys, 37 (2017), 1253-1290.  doi: 10.1017/etds.2015.82.  Google Scholar

[12]

W. Krieger and K. Matsumoto, Zeta functions and topological entropy of the Markov-Dyck shifts, Münster J. Math., 4 (2011), 171-184.   Google Scholar

[13]

W. Krieger and K. Matsumoto, A notion of synchronization of symbolic dynamics and a class of C*-algebras, Acta Appl. Math., 126 (2013), 263-275.  doi: 10.1007/s10440-013-9817-4.  Google Scholar

[14]

M. V. Lawson, Inverse Semigroups, World Scientific, Sigapure, New Jersey, London and Hong Kong, 1998. doi: 10.1142/9789812816689.  Google Scholar

[15]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  Google Scholar

[16]

K. Matsumoto, C*-algebras arising from Dyck systems of topological Markov chains, Math. Scand., 109 (2011), 31-54.  doi: 10.7146/math.scand.a-15176.  Google Scholar

[17]

K. Matsumoto, A certain synchronizing property of subshifts and flow equivalence, Israel J. Math., 196 (2013), 235-272.  doi: 10.1007/s11856-012-0159-0.  Google Scholar

[18]

M. Nivat and J.-F. Perrot, Une généralisation du monoîde bicyclique, C. R. Acad. Sc. Paris, 271 (1970), 824-827.   Google Scholar

[19]

D. Perrin, Algebraic combinatorics on words, Algebraic Combinatorics and Computer Science, H.Crapo, G.-C.Rota, Eds. Springer, 2001, 391-427.  Google Scholar

show all references

References:
[1]

A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shifts, Ergod. Th. & Dynam. Sys., 36 (2016), 470-513.  doi: 10.1017/etds.2014.74.  Google Scholar

[2]

G. Gordon and E. McMahon, A greedoid polynomial which distinguishes rooted arborescences, Proc. AMS., 107 (1989), 287-298.  doi: 10.1090/S0002-9939-1989-0967486-0.  Google Scholar

[3]

T. Hamachi and K. Inoue, Embeddings of shifts of finite type into the Dyck shift, Monatsh. Math., 145 (2005), 107-129.  doi: 10.1007/s00605-004-0297-5.  Google Scholar

[4]

T. Hamachi, K. Inoue and W. Krieger, Subsystems of finite type and semigroup invariants of subshifts, J. reine angew. Math., 632 (2009), 37-61.  doi: 10.1515/CRELLE.2009.049.  Google Scholar

[5]

T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups, arXiv: 1303.4158 [math.DS] Google Scholar

[6]

B. P. Kitchens, Symbolic Dynamics, Springer, Berlin, Heidelberg, New York, 1998.  Google Scholar

[7]

W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104.  doi: 10.1007/BF01762180.  Google Scholar

[8]

W. Krieger, On a syntactically defined invariant of symbolic dynamics, Ergod. Th. & Dynam.Sys, 10 (2000), 501-506.  doi: 10.1017/S0143385700000249.  Google Scholar

[9]

W. Krieger, On subshifts and semigroups, Bull. London Math. Soc., 38 (2006), 617-624.  doi: 10.1112/S0024609306018625.  Google Scholar

[10]

W. Krieger, On flow equivalence of R-graph shifts, Münster J. Math., 8 (2015), 229-239.   Google Scholar

[11]

W. Krieger, On subshift presentations, Ergod. Th. & Dynam. Sys, 37 (2017), 1253-1290.  doi: 10.1017/etds.2015.82.  Google Scholar

[12]

W. Krieger and K. Matsumoto, Zeta functions and topological entropy of the Markov-Dyck shifts, Münster J. Math., 4 (2011), 171-184.   Google Scholar

[13]

W. Krieger and K. Matsumoto, A notion of synchronization of symbolic dynamics and a class of C*-algebras, Acta Appl. Math., 126 (2013), 263-275.  doi: 10.1007/s10440-013-9817-4.  Google Scholar

[14]

M. V. Lawson, Inverse Semigroups, World Scientific, Sigapure, New Jersey, London and Hong Kong, 1998. doi: 10.1142/9789812816689.  Google Scholar

[15]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  Google Scholar

[16]

K. Matsumoto, C*-algebras arising from Dyck systems of topological Markov chains, Math. Scand., 109 (2011), 31-54.  doi: 10.7146/math.scand.a-15176.  Google Scholar

[17]

K. Matsumoto, A certain synchronizing property of subshifts and flow equivalence, Israel J. Math., 196 (2013), 235-272.  doi: 10.1007/s11856-012-0159-0.  Google Scholar

[18]

M. Nivat and J.-F. Perrot, Une généralisation du monoîde bicyclique, C. R. Acad. Sc. Paris, 271 (1970), 824-827.   Google Scholar

[19]

D. Perrin, Algebraic combinatorics on words, Algebraic Combinatorics and Computer Science, H.Crapo, G.-C.Rota, Eds. Springer, 2001, 391-427.  Google Scholar

Figure 1.  $G(1, 0, 2, 0, \dots)$
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Figure 2.  $G(1, (1, 0, 3, 0, \dots))$
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Figure 3.  $G_{2, 3}(4.1)$
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Figure 4.  $G_{3.2}(4.1)$
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