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January
2019, 39(1): 1-18.
doi: 10.3934/dcds.2019001
Markov-Dyck shifts, neutral periodic points and topological conjugacy
| 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 |
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 and Continuous Dynamical Systems,
2019, 39
(1)
: 1-18.
doi: 10.3934/dcds.2019001
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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. |
| [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. |
| [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. |
| [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. |
| [5] |
T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups,
arXiv: 1303.4158 [math.DS] |
| [6] |
B. P. Kitchens, Symbolic Dynamics, Springer, Berlin, Heidelberg, New York, 1998. |
| [7] |
W. Krieger,
On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104.
doi: 10.1007/BF01762180. |
| [8] |
W. Krieger,
On a syntactically defined invariant of symbolic dynamics, Ergod. Th. & Dynam.Sys, 10 (2000), 501-506.
doi: 10.1017/S0143385700000249. |
| [9] |
W. Krieger,
On subshifts and semigroups, Bull. London Math. Soc., 38 (2006), 617-624.
doi: 10.1112/S0024609306018625. |
| [10] |
W. Krieger,
On flow equivalence of R-graph shifts, Münster J. Math., 8 (2015), 229-239.
|
| [11] |
W. Krieger,
On subshift presentations, Ergod. Th. & Dynam. Sys, 37 (2017), 1253-1290.
doi: 10.1017/etds.2015.82. |
| [12] |
W. Krieger and K. Matsumoto,
Zeta functions and topological entropy of the Markov-Dyck shifts, Münster J. Math., 4 (2011), 171-184.
|
| [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. |
| [14] |
M. V. Lawson, Inverse Semigroups, World Scientific, Sigapure, New Jersey, London and Hong
Kong, 1998.
doi: 10.1142/9789812816689. |
| [15] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge
University Press, Cambridge, 1995. |
| [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. |
| [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. |
| [18] |
M. Nivat and J.-F. Perrot,
Une généralisation du monoîde bicyclique, C. R. Acad. Sc. Paris, 271 (1970), 824-827.
|
| [19] |
D. Perrin, Algebraic combinatorics on words, Algebraic Combinatorics and Computer Science, H.Crapo, G.-C.Rota, Eds. Springer, 2001, 391-427. |
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. |
| [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. |
| [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. |
| [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. |
| [5] |
T. Hamachi and W. Krieger, A construction of subshifts and a class of semigroups,
arXiv: 1303.4158 [math.DS] |
| [6] |
B. P. Kitchens, Symbolic Dynamics, Springer, Berlin, Heidelberg, New York, 1998. |
| [7] |
W. Krieger,
On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104.
doi: 10.1007/BF01762180. |
| [8] |
W. Krieger,
On a syntactically defined invariant of symbolic dynamics, Ergod. Th. & Dynam.Sys, 10 (2000), 501-506.
doi: 10.1017/S0143385700000249. |
| [9] |
W. Krieger,
On subshifts and semigroups, Bull. London Math. Soc., 38 (2006), 617-624.
doi: 10.1112/S0024609306018625. |
| [10] |
W. Krieger,
On flow equivalence of R-graph shifts, Münster J. Math., 8 (2015), 229-239.
|
| [11] |
W. Krieger,
On subshift presentations, Ergod. Th. & Dynam. Sys, 37 (2017), 1253-1290.
doi: 10.1017/etds.2015.82. |
| [12] |
W. Krieger and K. Matsumoto,
Zeta functions and topological entropy of the Markov-Dyck shifts, Münster J. Math., 4 (2011), 171-184.
|
| [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. |
| [14] |
M. V. Lawson, Inverse Semigroups, World Scientific, Sigapure, New Jersey, London and Hong
Kong, 1998.
doi: 10.1142/9789812816689. |
| [15] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge
University Press, Cambridge, 1995. |
| [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. |
| [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. |
| [18] |
M. Nivat and J.-F. Perrot,
Une généralisation du monoîde bicyclique, C. R. Acad. Sc. Paris, 271 (1970), 824-827.
|
| [19] |
D. Perrin, Algebraic combinatorics on words, Algebraic Combinatorics and Computer Science, H.Crapo, G.-C.Rota, Eds. Springer, 2001, 391-427. |
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