Continuous shift commuting maps between ultragraph shift spaces

Daniel Gonçalves; Marcelo Sobottka

doi:10.3934/dcds.2019043

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2019, Volume 39, Issue 2: 1033-1048. Doi: 10.3934/dcds.2019043

This issue Previous Article Intermediate Lyapunov exponents for systems with periodic orbit gluing property Next Article Pathological center foliation with dimension greater than one

Continuous shift commuting maps between ultragraph shift spaces

Daniel Gonçalves and
Marcelo Sobottka^,

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

* Corresponding author: marcelo.sobottka@ufsc.br
* Corresponding author: marcelo.sobottka@ufsc.br

Received: March 2018

Revised: August 2018

Published: February 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B10, 54H20; Secondary: 37B15.

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[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. Gonçalves, M. 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. Gonçalves, M. 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. Gonçalves, M. 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. Katsura, P. S. Muhly, A. 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.