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October  2020, 40(10): 5897-5909. doi: 10.3934/dcds.2020251

Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras

Kengo Matsumoto

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

Received  November 2019 Revised  May 2020 Published  June 2020

Fund Project: This work was supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537

We study KMS states for gauge actions with potential functions on Cuntz–Krieger algebras whose underlying one-sided topological Markov shifts are continuously orbit equivalent. As a result, we have a certain relationship between topological entropy of continuously orbit equivalent one-sided topological Markov shifts.

Keywords: Topological Markov shifts, orbit equivalence, Cuntz–Krieger algebras, KMS states, topological entropy.
Mathematics Subject Classification: Primary: 46L55; Secondary: 46L35, 37B10.
Citation: Kengo Matsumoto. Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5897-5909. doi: 10.3934/dcds.2020251
References:
[1]

V. Baladi, Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems, 18 (1998), 255-292.  doi: 10.1017/S0143385798113925.  Google Scholar

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer-Verlag, Berlin, 2008.  Google Scholar

[3]

O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag, New York, Berlin, Heidelberg, 1981. Google Scholar

[4]

K. A. Brix and T. M. Carlsen, Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids, Journal of the Australian Mathematical Society, 2019, arXiv: 1712.00179 [mathOA]. doi: 10.1017/S1446788719000168.  Google Scholar

[5]

T. M. Carlsen, S. Eilers, E. 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

[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]

M. Enomoto, M. Fujii and Y. Watatani, KMS states for gauge action on $O_{A}$, Math. Japon., 29 (1984), 607-619.   Google Scholar

[8]

R. Exel, Crossed products by finite index endomorphisms and KMS states, J. Funct. Anal., 199 (2003), 153-188.  doi: 10.1016/S0022-1236(02)00023-X.  Google Scholar

[9]

R. Exel, KMS states for generalized gauge actions on Cuntz–Krieger algebras, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 1-12.  doi: 10.1007/s00574-004-0001-3.  Google Scholar

[10]

R. Exel and A. O. Lopes, $C^*$-algebras and thermodynamic formalism, São Paulo J. Math. Sci., 2 (2008), 285-307.  doi: 10.11606/issn.2316-9028.v2i2p285-307.  Google Scholar

[11] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[12]

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

[13]

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

[14]

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

[15]

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

[18]

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

[19]

D. Olesen and G. K. Pedersen, Some $C^* $-dynamical systems with a single KMS state, Math. Scand., 42 (1978), 111-118.  doi: 10.7146/math.scand.a-11740.  Google Scholar

[20]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.  doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[21]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268pp.  Google Scholar

[22]

G. Pedersen, C*-Algebras and Their Automorphism Groups, London Mathematical Society Monographs, 14. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.  Google Scholar

[23]

C. Pinzari, Y. Watatani and K. Yonetani, KMS states, entropy and the variational principle in full $C^*$-dynamical systems, Comm. Math. Phys., 213 (2000), 331-379.  doi: 10.1007/s002200000244.  Google Scholar

[24]

D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978.  Google Scholar

[25]

D. Ruelle, Dynamical zeta functions and transfer operators, Notice Amer. Math. Soc., 49 (2002), 887-895.   Google Scholar

show all references

References:
[1]

V. Baladi, Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems, 18 (1998), 255-292.  doi: 10.1017/S0143385798113925.  Google Scholar

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer-Verlag, Berlin, 2008.  Google Scholar

[3]

O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag, New York, Berlin, Heidelberg, 1981. Google Scholar

[4]

K. A. Brix and T. M. Carlsen, Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids, Journal of the Australian Mathematical Society, 2019, arXiv: 1712.00179 [mathOA]. doi: 10.1017/S1446788719000168.  Google Scholar

[5]

T. M. Carlsen, S. Eilers, E. 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

[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]

M. Enomoto, M. Fujii and Y. Watatani, KMS states for gauge action on $O_{A}$, Math. Japon., 29 (1984), 607-619.   Google Scholar

[8]

R. Exel, Crossed products by finite index endomorphisms and KMS states, J. Funct. Anal., 199 (2003), 153-188.  doi: 10.1016/S0022-1236(02)00023-X.  Google Scholar

[9]

R. Exel, KMS states for generalized gauge actions on Cuntz–Krieger algebras, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 1-12.  doi: 10.1007/s00574-004-0001-3.  Google Scholar

[10]

R. Exel and A. O. Lopes, $C^*$-algebras and thermodynamic formalism, São Paulo J. Math. Sci., 2 (2008), 285-307.  doi: 10.11606/issn.2316-9028.v2i2p285-307.  Google Scholar

[11] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[12]

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

[13]

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

[14]

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

[15]

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

[18]

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

[19]

D. Olesen and G. K. Pedersen, Some $C^* $-dynamical systems with a single KMS state, Math. Scand., 42 (1978), 111-118.  doi: 10.7146/math.scand.a-11740.  Google Scholar

[20]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.  doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[21]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268pp.  Google Scholar

[22]

G. Pedersen, C*-Algebras and Their Automorphism Groups, London Mathematical Society Monographs, 14. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.  Google Scholar

[23]

C. Pinzari, Y. Watatani and K. Yonetani, KMS states, entropy and the variational principle in full $C^*$-dynamical systems, Comm. Math. Phys., 213 (2000), 331-379.  doi: 10.1007/s002200000244.  Google Scholar

[24]

D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978.  Google Scholar

[25]

D. Ruelle, Dynamical zeta functions and transfer operators, Notice Amer. Math. Soc., 49 (2002), 887-895.   Google Scholar

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