On the Markov-Dyck shifts of vertex type

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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

Abstract
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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 and Continuous Dynamical Systems, 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, ().
[2]	A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shifts, Ergodic Theory and Dynamical Systems, 74 (2014), 44pp, arXiv:1304.3487. doi: 10.1017/etds.2014.74.
[3]	J. Cuntz, Simple $C^$-algebras generated by isometries, Commun. Math. Phys.*, 57 (1977), 173-185. doi: 10.1007/BF01625776.
[4]	J. Cuntz and W. Krieger, A class of $C^$-algebras and topological Markov chains, Inventions Math.*, 56 (1980), 251-268. doi: 10.1007/BF01390048.
[5]	E. Deutsch, Dyck path enumeration, Discrete Math., 204 (1999), 167-202. doi: 10.1016/S0012-365X(98)00371-9.
[6]	I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley, New York, 1983.
[7]	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.
[8]	T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups,, preprint, ().
[9]	F. Harry, Line graphs, in Graph Theory, Massachusetts, Addison-Wesley, (1972), 71-83.
[10]	G. Keller, Circular codes, loop counting, and zeta-functions, J. Combinatorial Theory, 56 (1991), 75-83. doi: 10.1016/0097-3165(91)90023-A.
[11]	B. P. Kitchens, Symbolic Dynamics, Springer-Verlag, Berlin, Heidelberg and New York, 1998. doi: 10.1007/978-3-642-58822-8.
[12]	W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104. doi: 10.1007/BF01762180.
[13]	W. Krieger, On a syntactically defined invariant of symbolic dynamics, Ergodic Theory Dynam. Systems, 20 (2000), 501-516. doi: 10.1017/S0143385700000249.
[14]	W. Krieger, On subshifts and semigroups, Bull. London Math., 38 (2006), 617-624. doi: 10.1112/S0024609306018625.
[15]	W. Krieger and K. Matsumoto, Zeta functions and topological entropy of the Markov Dyck shifts, Münster J. Math., 4 (2011), 171-183.
[16]	W. Krieger and K. Matsumoto, Markov-Dyck shifts, neutral periodic points and topological conjugacy (tentative title),, in preparation., ().
[17]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[18]	K. Matsumoto, Cuntz-Krieger algebras and a generalization of Catalan numbers, Int. J. Math., 24 (2013), 1350040, 31pp. doi: 10.1142/S0129167X13500407.
[19]	K. Matsumoto, $C^$-algebras arising from Dyck systems of topological Markov chains, Math. Scand., 109* (2011), 31-54.
[20]	R. P. Stanley, Enumerative Combinatrics I, Wadsworth & Brooks/Cole Advanced Books, Monterey, CA, 1986. doi: 10.1007/978-1-4615-9763-6.

show all references

References:

[1]	M.-P. Béal, M. Blockelet and C. Dima, Sofic-Dick shifts,, preprint, ().
[2]	A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shifts, Ergodic Theory and Dynamical Systems, 74 (2014), 44pp, arXiv:1304.3487. doi: 10.1017/etds.2014.74.
[3]	J. Cuntz, Simple $C^$-algebras generated by isometries, Commun. Math. Phys.*, 57 (1977), 173-185. doi: 10.1007/BF01625776.
[4]	J. Cuntz and W. Krieger, A class of $C^$-algebras and topological Markov chains, Inventions Math.*, 56 (1980), 251-268. doi: 10.1007/BF01390048.
[5]	E. Deutsch, Dyck path enumeration, Discrete Math., 204 (1999), 167-202. doi: 10.1016/S0012-365X(98)00371-9.
[6]	I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley, New York, 1983.
[7]	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.
[8]	T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups,, preprint, ().
[9]	F. Harry, Line graphs, in Graph Theory, Massachusetts, Addison-Wesley, (1972), 71-83.
[10]	G. Keller, Circular codes, loop counting, and zeta-functions, J. Combinatorial Theory, 56 (1991), 75-83. doi: 10.1016/0097-3165(91)90023-A.
[11]	B. P. Kitchens, Symbolic Dynamics, Springer-Verlag, Berlin, Heidelberg and New York, 1998. doi: 10.1007/978-3-642-58822-8.
[12]	W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104. doi: 10.1007/BF01762180.
[13]	W. Krieger, On a syntactically defined invariant of symbolic dynamics, Ergodic Theory Dynam. Systems, 20 (2000), 501-516. doi: 10.1017/S0143385700000249.
[14]	W. Krieger, On subshifts and semigroups, Bull. London Math., 38 (2006), 617-624. doi: 10.1112/S0024609306018625.
[15]	W. Krieger and K. Matsumoto, Zeta functions and topological entropy of the Markov Dyck shifts, Münster J. Math., 4 (2011), 171-183.
[16]	W. Krieger and K. Matsumoto, Markov-Dyck shifts, neutral periodic points and topological conjugacy (tentative title),, in preparation., ().
[17]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[18]	K. Matsumoto, Cuntz-Krieger algebras and a generalization of Catalan numbers, Int. J. Math., 24 (2013), 1350040, 31pp. doi: 10.1142/S0129167X13500407.
[19]	K. Matsumoto, $C^$-algebras arising from Dyck systems of topological Markov chains, Math. Scand., 109* (2011), 31-54.
[20]	R. P. Stanley, Enumerative Combinatrics I, Wadsworth & Brooks/Cole Advanced Books, Monterey, CA, 1986. doi: 10.1007/978-1-4615-9763-6.

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