February 2019, 39(2): 1033-1048. doi: 10.3934/dcds.2019043

Continuous shift commuting maps between ultragraph shift spaces

UFSC – Department of Mathematics, 88040-900 Florianópolis - SC, Brazil

* Corresponding author: marcelo.sobottka@ufsc.br

Received  March 2018 Revised  August 2018 Published  November 2018

Fund Project: D. Gonçalves was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq. M. Sobottka was supported by CNPq-Brazil PQ grant

Recently a generalization of shifts of finite type to the infinite alphabet case was proposed, in connection with the theory of ultragraph C*-algebras. In this work we characterize the class of continuous shift commuting maps between these spaces. In particular, we prove a Curtis-Hedlund-Lyndon type theorem and use it to completely characterize continuous, shift commuting, length preserving maps in terms of generalized sliding block codes.

Citation: Daniel Gonçalves, Marcelo Sobottka. Continuous shift commuting maps between ultragraph shift spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1033-1048. doi: 10.3934/dcds.2019043
References:
[1]

T. M. Carlsen, E. Ruiz, A. Sims and M. Tomforde, Reconstruction of groupoids and C*-rigidity of dynamical systems, preprint, arXiv: 1711.01052.

[2]

G. G. Castro and D. Gonçalves, KMS and ground states on ultragraph C*-algebras, Integral Equations and Operator Theory, 90 (2018), 63. doi: 10.1007/s00020-018-2490-2.

[3]

T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, 2010. doi: 10.1007/978-3-642-14034-1.

[4]

J. Cuntz and W. Krieger, A class of C*-algebras and topological Markov chains, Invent. Math., 56 (1980), 251-268. doi: 10.1007/BF01390048.

[5]

D. Fiebig, Factor maps, entropy and fiber cardinality for Markov shifts, Rocky Mountain J. Math., 31 (2001), 955-986. doi: 10.1216/rmjm/1020171674.

[6]

D. Fiebig, Graphs with pre-assigned Salama entropies and optimal degress, Ergodic Theory Dynam. Systems, 23 (2003), 1093-1124. doi: 10.1017/S014338570200161X.

[7]

D. Fiebig and U.-R. Fiebig, Topological boundaries for countable state Markov shifts, Proc. London Math. Soc., 70 (1995), 625-643. doi: 10.1112/plms/s3-70.3.625.

[8]

D. Fiebig and U.-R. Fiebig, Embedding theorems for locally compact Markov shifts, Ergodic Theory Dynam. Systems, 25 (2005), 107-131. doi: 10.1017/S0143385704000689.

[9]

D. Gonçalves, H. Li and D. Royer, Branching systems and general Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras, Internat. J. Math., 27 (2016), 1650083, 26pp. doi: 10.1142/S0129167X1650083X.

[10]

D. Gonçalves and D. Royer, (M+1)-step shift spaces that are not conjugate to M-step shift spaces, Bull. Sci. Math., 139 (2015), 178-183. doi: 10.1016/j.bulsci.2014.08.007.

[11]

D. Gonçalves and D. Royer, Ultragraphs and shift spaces over infinite alphabets, Bull. Sci. Math., 141 (2017), 25-45. doi: 10.1016/j.bulsci.2016.10.002.

[12]

D. Gonçalves and D. Royer, Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras, Int. Math. Res. Not., rnx175. doi: 10.1093/imrn/rnx175.

[13]

D. GonçalvesM. Sobottka and C. Starling, Sliding block codes between shift spaces over infinite alphabets, Math. Nachr., 289 (2016), 2178-2191. doi: 10.1002/mana.201500309.

[14]

D. GonçalvesM. Sobottka and C. Starling, Two-sided shift spaces over infinite alphabets, J. Aust. Math. Soc., 103 (2017), 357-386. doi: 10.1017/S1446788717000039.

[15]

D. GonçalvesM. Sobottka and C. Starling, Inverse semigroup shifts over countable alphabets, Semigroup Forum, 96 (2018), 203-240. doi: 10.1007/s00233-017-9858-5.

[16]

D. Gonçalves and B. B. Uggioni, Li-Yorke chaos for ultragraph shift spaces, preprint, arXiv: 1806.07927.

[17]

T. KatsuraP. S. MuhlyA. Sims and M. Tomforde, Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence, J. Reine Angew. Math., 640 (2010), 135-165. doi: 10.1515/CRELLE.2010.023.

[18]

B. P. Kitchens, Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts, Springer-Verlag, 1998. doi: 10.1007/978-3-642-58822-8.

[19]

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

[20]

A. Marrero and P. S. Muhly, Groupoid and inverse semigroup presentations of ultragraph C*-algebras, Semigroup Forum, 77 (2008), 399-422. doi: 10.1007/s00233-008-9046-8.

[21]

W. Ott, M. Tomforde and P. N. Willis, One-sided shift spaces over infinite alphabets, New York J. Math., NYJM Monographs, 5 (2014), 54 pp.

[22]

K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448. doi: 10.1017/S014338570000359X.

[23]

I. A. Salama, Topological entropy and recurrence of countable chains, Pacific J. Math., 134 (1988), 325-341.

[24]

M. Sobottka and D. Gonçalves, A note on the definition of sliding block codes and the Curtis-Hedlund-Lyndon Theorem, J. Cell. Autom., 12 (2017), 209-215.

[25]

M. Tomforde, A unified approach to Exel-Laca algebras and C*-algebras associated to graphs, J. Operator Theory, 50 (2003), 345-368.

[26]

M. Tomforde, Simplicity of ultragraph algebras, Indiana Univ. Math. J., 52 (2003), 901-925. doi: 10.1512/iumj.2003.52.2209.

[27]

J. von Neumann, Theory of Self-reproducing Automata, (edited and completed by A. W. Burks), University of Illinois Press, 1966.

[28]

S. B. G. Webster, The path space of a directed graph, Proc. Amer. Math. Soc., 142 (2014), 213-225. doi: 10.1090/S0002-9939-2013-11755-7.

show all references

References:
[1]

T. M. Carlsen, E. Ruiz, A. Sims and M. Tomforde, Reconstruction of groupoids and C*-rigidity of dynamical systems, preprint, arXiv: 1711.01052.

[2]

G. G. Castro and D. Gonçalves, KMS and ground states on ultragraph C*-algebras, Integral Equations and Operator Theory, 90 (2018), 63. doi: 10.1007/s00020-018-2490-2.

[3]

T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, 2010. doi: 10.1007/978-3-642-14034-1.

[4]

J. Cuntz and W. Krieger, A class of C*-algebras and topological Markov chains, Invent. Math., 56 (1980), 251-268. doi: 10.1007/BF01390048.

[5]

D. Fiebig, Factor maps, entropy and fiber cardinality for Markov shifts, Rocky Mountain J. Math., 31 (2001), 955-986. doi: 10.1216/rmjm/1020171674.

[6]

D. Fiebig, Graphs with pre-assigned Salama entropies and optimal degress, Ergodic Theory Dynam. Systems, 23 (2003), 1093-1124. doi: 10.1017/S014338570200161X.

[7]

D. Fiebig and U.-R. Fiebig, Topological boundaries for countable state Markov shifts, Proc. London Math. Soc., 70 (1995), 625-643. doi: 10.1112/plms/s3-70.3.625.

[8]

D. Fiebig and U.-R. Fiebig, Embedding theorems for locally compact Markov shifts, Ergodic Theory Dynam. Systems, 25 (2005), 107-131. doi: 10.1017/S0143385704000689.

[9]

D. Gonçalves, H. Li and D. Royer, Branching systems and general Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras, Internat. J. Math., 27 (2016), 1650083, 26pp. doi: 10.1142/S0129167X1650083X.

[10]

D. Gonçalves and D. Royer, (M+1)-step shift spaces that are not conjugate to M-step shift spaces, Bull. Sci. Math., 139 (2015), 178-183. doi: 10.1016/j.bulsci.2014.08.007.

[11]

D. Gonçalves and D. Royer, Ultragraphs and shift spaces over infinite alphabets, Bull. Sci. Math., 141 (2017), 25-45. doi: 10.1016/j.bulsci.2016.10.002.

[12]

D. Gonçalves and D. Royer, Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras, Int. Math. Res. Not., rnx175. doi: 10.1093/imrn/rnx175.

[13]

D. GonçalvesM. Sobottka and C. Starling, Sliding block codes between shift spaces over infinite alphabets, Math. Nachr., 289 (2016), 2178-2191. doi: 10.1002/mana.201500309.

[14]

D. GonçalvesM. Sobottka and C. Starling, Two-sided shift spaces over infinite alphabets, J. Aust. Math. Soc., 103 (2017), 357-386. doi: 10.1017/S1446788717000039.

[15]

D. GonçalvesM. Sobottka and C. Starling, Inverse semigroup shifts over countable alphabets, Semigroup Forum, 96 (2018), 203-240. doi: 10.1007/s00233-017-9858-5.

[16]

D. Gonçalves and B. B. Uggioni, Li-Yorke chaos for ultragraph shift spaces, preprint, arXiv: 1806.07927.

[17]

T. KatsuraP. S. MuhlyA. Sims and M. Tomforde, Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence, J. Reine Angew. Math., 640 (2010), 135-165. doi: 10.1515/CRELLE.2010.023.

[18]

B. P. Kitchens, Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts, Springer-Verlag, 1998. doi: 10.1007/978-3-642-58822-8.

[19]

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

[20]

A. Marrero and P. S. Muhly, Groupoid and inverse semigroup presentations of ultragraph C*-algebras, Semigroup Forum, 77 (2008), 399-422. doi: 10.1007/s00233-008-9046-8.

[21]

W. Ott, M. Tomforde and P. N. Willis, One-sided shift spaces over infinite alphabets, New York J. Math., NYJM Monographs, 5 (2014), 54 pp.

[22]

K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448. doi: 10.1017/S014338570000359X.

[23]

I. A. Salama, Topological entropy and recurrence of countable chains, Pacific J. Math., 134 (1988), 325-341.

[24]

M. Sobottka and D. Gonçalves, A note on the definition of sliding block codes and the Curtis-Hedlund-Lyndon Theorem, J. Cell. Autom., 12 (2017), 209-215.

[25]

M. Tomforde, A unified approach to Exel-Laca algebras and C*-algebras associated to graphs, J. Operator Theory, 50 (2003), 345-368.

[26]

M. Tomforde, Simplicity of ultragraph algebras, Indiana Univ. Math. J., 52 (2003), 901-925. doi: 10.1512/iumj.2003.52.2209.

[27]

J. von Neumann, Theory of Self-reproducing Automata, (edited and completed by A. W. Burks), University of Illinois Press, 1966.

[28]

S. B. G. Webster, The path space of a directed graph, Proc. Amer. Math. Soc., 142 (2014), 213-225. doi: 10.1090/S0002-9939-2013-11755-7.

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