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January  2016, 36(1): 403-422. doi: 10.3934/dcds.2016.36.403

On the Markov-Dyck shifts of vertex type

Kengo Matsumoto 1,

1. 

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

Received  May 2014 Revised  April 2015 Published  June 2015

For a given finite directed graph $G$, there are two types of Markov-Dyck shifts, the Markov-Dyck shift $D_G^V$ of vertex type and the Markov-Dyck shift $D_G^E$ of edge type. It is shown that, if $G$ does not have multi-edges, the former is a finite-to-one factor of the latter, and they have the same topological entropy. An expression for the zeta function of a Markov-Dyck shift of vertex type is given. It is different from that of the Markov-Dyck shift of edge type.
Keywords: Catalan numbers., zeta function, subshift, Markov-Dyck shift, entropy.
Mathematics Subject Classification: Primary: 37B10; Secondary: 46L05, 05A1.
Citation: 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
References:
[1]

M.-P. Béal, M. Blockelet and C. Dima, Sofic-Dick shifts,, preprint, ().   Google Scholar

[2]

A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shifts,, Ergodic Theory and Dynamical Systems, 74 (2014).  doi: 10.1017/etds.2014.74.  Google Scholar

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J. Cuntz, Simple $C^*$-algebras generated by isometries,, Commun. Math. Phys., 57 (1977), 173.  doi: 10.1007/BF01625776.  Google Scholar

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J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains,, Inventions Math., 56 (1980), 251.  doi: 10.1007/BF01390048.  Google Scholar

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E. Deutsch, Dyck path enumeration,, Discrete Math., 204 (1999), 167.  doi: 10.1016/S0012-365X(98)00371-9.  Google Scholar

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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration,, John Wiley, (1983).   Google Scholar

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

[8]

T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups,, preprint, ().   Google Scholar

[9]

F. Harry, Line graphs, in, Graph Theory, (1972), 71.   Google Scholar

[10]

G. Keller, Circular codes, loop counting, and zeta-functions,, J. Combinatorial Theory, 56 (1991), 75.  doi: 10.1016/0097-3165(91)90023-A.  Google Scholar

[11]

B. P. Kitchens, Symbolic Dynamics,, Springer-Verlag, (1998).  doi: 10.1007/978-3-642-58822-8.  Google Scholar

[12]

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

[13]

W. Krieger, On a syntactically defined invariant of symbolic dynamics,, Ergodic Theory Dynam. Systems, 20 (2000), 501.  doi: 10.1017/S0143385700000249.  Google Scholar

[14]

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

[15]

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

[16]

W. Krieger and K. Matsumoto, Markov-Dyck shifts, neutral periodic points and topological conjugacy (tentative title),, in preparation., ().   Google Scholar

[17]

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

[18]

K. Matsumoto, Cuntz-Krieger algebras and a generalization of Catalan numbers,, Int. J. Math., 24 (2013).  doi: 10.1142/S0129167X13500407.  Google Scholar

[19]

K. Matsumoto, $C^*$-algebras arising from Dyck systems of topological Markov chains,, Math. Scand., 109 (2011), 31.   Google Scholar

[20]

R. P. Stanley, Enumerative Combinatrics I,, Wadsworth & Brooks/Cole Advanced Books, (1986).  doi: 10.1007/978-1-4615-9763-6.  Google Scholar

show all references

References:
[1]

M.-P. Béal, M. Blockelet and C. Dima, Sofic-Dick shifts,, preprint, ().   Google Scholar

[2]

A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shifts,, Ergodic Theory and Dynamical Systems, 74 (2014).  doi: 10.1017/etds.2014.74.  Google Scholar

[3]

J. Cuntz, Simple $C^*$-algebras generated by isometries,, Commun. Math. Phys., 57 (1977), 173.  doi: 10.1007/BF01625776.  Google Scholar

[4]

J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains,, Inventions Math., 56 (1980), 251.  doi: 10.1007/BF01390048.  Google Scholar

[5]

E. Deutsch, Dyck path enumeration,, Discrete Math., 204 (1999), 167.  doi: 10.1016/S0012-365X(98)00371-9.  Google Scholar

[6]

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration,, John Wiley, (1983).   Google Scholar

[7]

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

[8]

T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups,, preprint, ().   Google Scholar

[9]

F. Harry, Line graphs, in, Graph Theory, (1972), 71.   Google Scholar

[10]

G. Keller, Circular codes, loop counting, and zeta-functions,, J. Combinatorial Theory, 56 (1991), 75.  doi: 10.1016/0097-3165(91)90023-A.  Google Scholar

[11]

B. P. Kitchens, Symbolic Dynamics,, Springer-Verlag, (1998).  doi: 10.1007/978-3-642-58822-8.  Google Scholar

[12]

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

[13]

W. Krieger, On a syntactically defined invariant of symbolic dynamics,, Ergodic Theory Dynam. Systems, 20 (2000), 501.  doi: 10.1017/S0143385700000249.  Google Scholar

[14]

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

[15]

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

[16]

W. Krieger and K. Matsumoto, Markov-Dyck shifts, neutral periodic points and topological conjugacy (tentative title),, in preparation., ().   Google Scholar

[17]

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

[18]

K. Matsumoto, Cuntz-Krieger algebras and a generalization of Catalan numbers,, Int. J. Math., 24 (2013).  doi: 10.1142/S0129167X13500407.  Google Scholar

[19]

K. Matsumoto, $C^*$-algebras arising from Dyck systems of topological Markov chains,, Math. Scand., 109 (2011), 31.   Google Scholar

[20]

R. P. Stanley, Enumerative Combinatrics I,, Wadsworth & Brooks/Cole Advanced Books, (1986).  doi: 10.1007/978-1-4615-9763-6.  Google Scholar

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