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Extended symmetry groups of multidimensional subshifts with hierarchical structure
Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras
Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan |
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.
References:
[1] |
V. Baladi,
Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems, 18 (1998), 255-292.
doi: 10.1017/S0143385798113925. |
[2] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer-Verlag, Berlin, 2008. |
[3] |
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag, New York, Berlin, Heidelberg, 1981. |
[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. |
[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. |
[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] |
M. Enomoto, M. Fujii and Y. Watatani,
KMS states for gauge action on $O_{A}$, Math. Japon., 29 (1984), 607-619.
|
[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. |
[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. |
[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. |
[11] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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 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. |
[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. |
[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. |
[20] |
W. Parry,
Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.
doi: 10.1090/S0002-9947-1964-0161372-1. |
[21] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268pp. |
[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. |
[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. |
[24] |
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978. |
[25] |
D. Ruelle,
Dynamical zeta functions and transfer operators, Notice Amer. Math. Soc., 49 (2002), 887-895.
|
show all references
References:
[1] |
V. Baladi,
Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems, 18 (1998), 255-292.
doi: 10.1017/S0143385798113925. |
[2] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer-Verlag, Berlin, 2008. |
[3] |
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag, New York, Berlin, Heidelberg, 1981. |
[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. |
[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. |
[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] |
M. Enomoto, M. Fujii and Y. Watatani,
KMS states for gauge action on $O_{A}$, Math. Japon., 29 (1984), 607-619.
|
[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. |
[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. |
[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. |
[11] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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 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. |
[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. |
[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. |
[20] |
W. Parry,
Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.
doi: 10.1090/S0002-9947-1964-0161372-1. |
[21] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268pp. |
[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. |
[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. |
[24] |
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978. |
[25] |
D. Ruelle,
Dynamical zeta functions and transfer operators, Notice Amer. Math. Soc., 49 (2002), 887-895.
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