doi: 10.3934/dcds.2021139
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Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

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

Received  April 2021 Revised  August 2021 Early access September 2021

Fund Project: The author is supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537

We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.

Citation: Kengo Matsumoto. Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021139
References:
[1]

M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210.  doi: 10.1007/BF02761039.  Google Scholar

[2]

M. Boyle, B. Marcus and P. Trow, Resolving maps and the dimension group for shifts of finite type, Mem. Amer. Math. Soc., 70 (1987) no. 377,146 pp. doi: 10.1090/memo/0377.  Google Scholar

[3]

K. A. Brix and T. M. Carlsen, Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids, J. Aust. Math. Soc., 109 (2020), 289-298.  doi: 10.1017/S1446788719000168.  Google Scholar

[4]

T. M. CarlsenS. EilersE. Ortega and G. Restorff, Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids, J. Math. Anal. Appl., 469 (2019), 1088-1110.  doi: 10.1016/j.jmaa.2018.09.056.  Google Scholar

[5]

T. M. CarlsenE. Ruiz and A. Sims, Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras, Proc. Amer. Math. Soc., 145 (2017), 1581-1592.  doi: 10.1090/proc/13321.  Google Scholar

[6]

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

[7]

T. GiordanoI. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel. J. Math., 111 (1999), 285-320.  doi: 10.1007/BF02810689.  Google Scholar

[8]

B. P. Kitchens, Symbolic Dynamics, Springer-Verlag, Berlin, Heidelberg and New York, 1998. doi: 10.1007/978-3-642-58822-8.  Google Scholar

[9] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[10]

K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math., 246 (2010), 199-225.  doi: 10.2140/pjm.2010.246.199.  Google Scholar

[11]

K. Matsumoto, K-groups of the full group actions on one-sided topological Markov shifts, Discrete and Contin. Dyn. Syst., 33 (2013), 3753-3765.  doi: 10.3934/dcds.2013.33.3753.  Google Scholar

[12]

K. Matsumoto, Classification of Cuntz–Krieger algebras by orbit equivalence of topological Markov shifts, Proc. Amer. Math. Soc., 141 (2013), 2329-2342.  doi: 10.1090/S0002-9939-2013-11519-4.  Google Scholar

[13]

K. Matsumoto, Full groups of one-sided topological Markov shifts, Israel J. Math., 205 (2015), 1-33.  doi: 10.1007/s11856-014-1134-8.  Google Scholar

[14]

K. Matsumoto, Strongly continuous orbit equivalence of one-sided topological Markov shifts, J. Operator Theory, 74 (2015), 457-483.  doi: 10.7900/jot.2014aug19.2063.  Google Scholar

[15]

K. Matsumoto, On flow equivalence of one-sided topological Markov shifts, Proc. Amer. Math. Soc., 144 (2016), 2923-2937.  doi: 10.1090/proc/13074.  Google Scholar

[16]

K. Matsumoto, Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras, Proc. Amer. Math. Soc., 145 (2017), 1131-1140.  doi: 10.1090/proc/13387.  Google Scholar

[17]

K. Matsumoto, Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras, Math. Z., 285 (2017), 121-141.  doi: 10.1007/s00209-016-1700-3.  Google Scholar

[18]

K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras, Kyoto J. Math., 54 (2014), 863-878.  doi: 10.1215/21562261-2801849.  Google Scholar

[19]

K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions, Ergodic Theory Dynam. Systems, 36 (2016), 1557-1581.  doi: 10.1017/etds.2014.128.  Google Scholar

[20]

K. Matsumoto and H. Matui, Full groups of Cuntz-Krieger algebras and Higman-Thompson groups, Groups Geom. Dyn., 11 (2017), 499-531.  doi: 10.4171/GGD/405.  Google Scholar

[21]

H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. London Math. Soc., 104 (2012), 27-56.  doi: 10.1112/plms/pdr029.  Google Scholar

[22]

H. Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84.  doi: 10.1515/crelle-2013-0041.  Google Scholar

[23]

B. Parry and D. Sullivan, A topological invariant for flows on one-dimensional spaces, Topology, 14 (1975), 297-299.  doi: 10.1016/0040-9383(75)90012-9.  Google Scholar

[24]

Y. T. Poon, A $K$-theoretic invariant for dynamical systems, Trans. Amer. Math. Soc., 311 (1989), 513-533.  doi: 10.2307/2001140.  Google Scholar

[25]

J. Tomiyama, Topological full groups and structure of normalizers in transformation group $C^*$-algebras, Pacific. J. Math., 173 (1996), 571-583.  doi: 10.2140/pjm.1996.173.571.  Google Scholar

show all references

References:
[1]

M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210.  doi: 10.1007/BF02761039.  Google Scholar

[2]

M. Boyle, B. Marcus and P. Trow, Resolving maps and the dimension group for shifts of finite type, Mem. Amer. Math. Soc., 70 (1987) no. 377,146 pp. doi: 10.1090/memo/0377.  Google Scholar

[3]

K. A. Brix and T. M. Carlsen, Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids, J. Aust. Math. Soc., 109 (2020), 289-298.  doi: 10.1017/S1446788719000168.  Google Scholar

[4]

T. M. CarlsenS. EilersE. Ortega and G. Restorff, Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids, J. Math. Anal. Appl., 469 (2019), 1088-1110.  doi: 10.1016/j.jmaa.2018.09.056.  Google Scholar

[5]

T. M. CarlsenE. Ruiz and A. Sims, Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras, Proc. Amer. Math. Soc., 145 (2017), 1581-1592.  doi: 10.1090/proc/13321.  Google Scholar

[6]

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

[7]

T. GiordanoI. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel. J. Math., 111 (1999), 285-320.  doi: 10.1007/BF02810689.  Google Scholar

[8]

B. P. Kitchens, Symbolic Dynamics, Springer-Verlag, Berlin, Heidelberg and New York, 1998. doi: 10.1007/978-3-642-58822-8.  Google Scholar

[9] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[10]

K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math., 246 (2010), 199-225.  doi: 10.2140/pjm.2010.246.199.  Google Scholar

[11]

K. Matsumoto, K-groups of the full group actions on one-sided topological Markov shifts, Discrete and Contin. Dyn. Syst., 33 (2013), 3753-3765.  doi: 10.3934/dcds.2013.33.3753.  Google Scholar

[12]

K. Matsumoto, Classification of Cuntz–Krieger algebras by orbit equivalence of topological Markov shifts, Proc. Amer. Math. Soc., 141 (2013), 2329-2342.  doi: 10.1090/S0002-9939-2013-11519-4.  Google Scholar

[13]

K. Matsumoto, Full groups of one-sided topological Markov shifts, Israel J. Math., 205 (2015), 1-33.  doi: 10.1007/s11856-014-1134-8.  Google Scholar

[14]

K. Matsumoto, Strongly continuous orbit equivalence of one-sided topological Markov shifts, J. Operator Theory, 74 (2015), 457-483.  doi: 10.7900/jot.2014aug19.2063.  Google Scholar

[15]

K. Matsumoto, On flow equivalence of one-sided topological Markov shifts, Proc. Amer. Math. Soc., 144 (2016), 2923-2937.  doi: 10.1090/proc/13074.  Google Scholar

[16]

K. Matsumoto, Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras, Proc. Amer. Math. Soc., 145 (2017), 1131-1140.  doi: 10.1090/proc/13387.  Google Scholar

[17]

K. Matsumoto, Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras, Math. Z., 285 (2017), 121-141.  doi: 10.1007/s00209-016-1700-3.  Google Scholar

[18]

K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras, Kyoto J. Math., 54 (2014), 863-878.  doi: 10.1215/21562261-2801849.  Google Scholar

[19]

K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions, Ergodic Theory Dynam. Systems, 36 (2016), 1557-1581.  doi: 10.1017/etds.2014.128.  Google Scholar

[20]

K. Matsumoto and H. Matui, Full groups of Cuntz-Krieger algebras and Higman-Thompson groups, Groups Geom. Dyn., 11 (2017), 499-531.  doi: 10.4171/GGD/405.  Google Scholar

[21]

H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. London Math. Soc., 104 (2012), 27-56.  doi: 10.1112/plms/pdr029.  Google Scholar

[22]

H. Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84.  doi: 10.1515/crelle-2013-0041.  Google Scholar

[23]

B. Parry and D. Sullivan, A topological invariant for flows on one-dimensional spaces, Topology, 14 (1975), 297-299.  doi: 10.1016/0040-9383(75)90012-9.  Google Scholar

[24]

Y. T. Poon, A $K$-theoretic invariant for dynamical systems, Trans. Amer. Math. Soc., 311 (1989), 513-533.  doi: 10.2307/2001140.  Google Scholar

[25]

J. Tomiyama, Topological full groups and structure of normalizers in transformation group $C^*$-algebras, Pacific. J. Math., 173 (1996), 571-583.  doi: 10.2140/pjm.1996.173.571.  Google Scholar

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