February  2022, 42(2): 841-862. doi: 10.3934/dcds.2021139

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 Published  February 2022 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 and Continuous Dynamical Systems, 2022, 42 (2) : 841-862. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

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