Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T04:38:24.523Z Has data issue: false hasContentIssue false
  • English
  • Français

KMS states and branched points

Published online by Cambridge University Press:  01 December 2007

MASAKI IZUMI ,
TSUYOSHI KAJIWARA  and
YASUO WATATANI
MASAKI IZUMI
Affiliation:
Department of Mathematics, Graduate School of Sciences, Kyoto University, Kyoto 606-8502, Japan (email: [email protected])
TSUYOSHI KAJIWARA
Affiliation:
Department of Environmental and Mathematical Sciences, Okayama University, Tsushima 700-8530, Japan (email: [email protected])
YASUO WATATANI
Affiliation:
Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We completely classify the Kubo–Martin–Schwinger (KMS) states for the gauge action on a C*-algebra associated with a rational function R introduced in our previous work. The gauge action has a phase transition at β=log deg R. We can recover the degree of R, the number of branched points, the number of exceptional points and the orbits of exceptional points from the structure of the KMS states. We also classify the KMS states for C*-algebras associated with some self-similar sets, including the full tent map and the Sierpinski gasket by a similar method.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1887 - 1918
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[2]Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics, 132). Springer, New York, 1991.CrossRefGoogle Scholar
[3]Bratteli, O.. Inductive limits of finite dimensional C *-algebras. Trans. Amer. Math. Soc. 171 (1972), 195–234.Google Scholar
[4]Bratteli, O. and Jorgensen, P. E. T.. Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N. Integral Equations Operator Theory 28 (1997), 382–443.CrossRefGoogle Scholar
[5]Bratteli, O. and Jorgensen, P. E. T.. Wavelets Through a Looking Glass: The World of the Spectrum. Birkhäuser, Basel, 2002.CrossRefGoogle Scholar
[6]Bratteli, O., Jorgensen, P. E. T., Kim, K. H. and Roush, F.. Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups. Ergod. Th. & Dynam. Sys. 21 (2001), 1625–1655.CrossRefGoogle Scholar
[7]Brenken, B.. C *-algebras associated with topological relations. J. Ramanujan Math. Soc. 19 (2004), 35–55.Google Scholar
[8]Cuntz, J.. Simple C *-algebras generated by isometries. Comm. Math. Phys. 57 (1977), 173–185.CrossRefGoogle Scholar
[9]Cuntz, J. and Krieger, W.. A class of C *-algebras and topological Markov chains. Invent. Math. 56 (1980), 251–268.CrossRefGoogle Scholar
[10]Deaconu, V. and Muhly, M.. C *-algebras associated with branched coverings. Proc. Amer. Math. Soc. 129 (2001), 1077–1086.CrossRefGoogle Scholar
[11]Deaconu, V. and Shultz, F.. C *-algebras associated with interval maps. Trans. Amer. Math. Soc. 359 (2007), 1889–1924.CrossRefGoogle Scholar
[12]Doyle, P. and McMullen, C.. Solving the quintic by iteration. Acta Math. 163 (1989), 151–180.CrossRefGoogle Scholar
[13]Dutkay, D. E. and Jorgensen, P. E. T.. Hilbert spaces built on a similarity and on dynamical renormalization. J. Math. Phys. 47(5) (2006), 20pp (053504).CrossRefGoogle Scholar
[14]Enomoto, M., Fujii, M. and Watatani, Y.. KMS states for gauge action on O A. Math. Japon. 29 (1984), 607–619.Google Scholar
[15]Evans, D.. On O n. Publ. Res. Inst. Math. Sci. Kyoto Univ. 16 (1980), 915–927.CrossRefGoogle Scholar
[16]Exel, R.. Crossed-products by finite index endomorphisms and KMS states. J. Funct. Anal. 199 (2003), 153–183.CrossRefGoogle Scholar
[17]Exel, R.. KMS states for generalized gauge actions on Cuntz–Krieger algebras (An application of the Ruelle–Perron–Frobenius theorem). Bull. Braz. Math. Soc. (N.S.) 35 (2004), 1–12.CrossRefGoogle Scholar
[18]Exel, R. and Laca, M.. Partial dynamical systems and the KMS condition. Comm. Math. Phys. 232 (2003), 223–277.CrossRefGoogle Scholar
[19]Fowler, N. J., Muhly, P. S. and Raeburn, I.. Representations of Cuntz–Pimsner algebras. Indiana Univ. Math. J. 52 (2003), 569–605.CrossRefGoogle Scholar
[20]Freire, A., Lopes, A. and Mäné, R.. An invariant measure for rational maps. Bol. Soc. Brasil. Mat. (N.S.) 14 (1983), 45–62.CrossRefGoogle Scholar
[21]Heicklen, D. and Hoffman, C.. Rational maps are d-adic Bernoulli. Ann. of Math. (2) 156 (2002), 103–114.CrossRefGoogle Scholar
[22]Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[23]Izumi, M.. Subalgebras of infinite C *-algebras with finite Watatani indices I, Cuntz algebras. Comm. Math. Phys. 155 (1993), 157–182.CrossRefGoogle Scholar
[24]Jorgensen, P. E. T.. Ruelle Operators: Functions Which Are Harmonic With Respect to a Transfer Operator (Memoirs of the American Mathematical Society, vol. 152, no. 720). American Mathematical Society, Providence, RI, 2001.Google Scholar
[25]Kajiwara, T., Pinzari, C. and Watatani, Y.. Jones index theory for Hilbert C *-bimodules and its equivalence with conjugation theory. J. Funct. Anal. 215 (2004), 1–49.CrossRefGoogle Scholar
[26]Kajiwara, T. and Watatani, Y.. C *-algebras associated with complex dynamical systems. Indiana Math. J. 54 (2005), 755–778.CrossRefGoogle Scholar
[27]Kajiwara, T. and Watatani, Y.. C *-algebras associated with self-similar sets. J. Operator Theory 56 (2006), 225–247.Google Scholar
[28]Kajiwara, T. and Watatani, Y.. KMS states on C *-algebras associated with self-similar sets. Preprint, math.OA/04055142004.Google Scholar
[29]Katayama, Y. and Takehana, H.. On automorphisms of generalized Cuntz algebras. Internat. J. Math. 9 (1998), 493–512.CrossRefGoogle Scholar
[30]Kerr, D. and Pinzari, C.. Noncommutative pressure and the variational principle in Cuntz–Krieger-type C *-algebras. J. Funct. Anal. 188 (2002), 156–215.CrossRefGoogle Scholar
[31]Kumjian, A. and Renault, J.. KMS states on C *-algebras associated to expansive maps. Proc. Amer. Math. Soc. 134 (2006), 2067–2078.CrossRefGoogle Scholar
[32]Laca, M. and Neshveyev, S.. KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211 (2004), 457–482.CrossRefGoogle Scholar
[33]Lyubich, M. Yu.. Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. & Dynam. Sys. 3 (1983), 351–385.CrossRefGoogle Scholar
[34]Matsumoto, K.. K-theoretic invariants and conformal measures of the Dyck shifts. Internat. J. Math. 16 (2005), 213–248.CrossRefGoogle Scholar
[35]Matsumoto, K., Watatani, Y. and Yoshida, M.. KMS-states for gauge action on C *-algebras associated with subshifts. Math. Z. 228 (1998), 489–509.CrossRefGoogle Scholar
[36]Muhly, P. and Solel, B.. On the Morita equivalence of tensor algebras. Proc. London Math. Soc. (3) 81 (2000), 113–168.CrossRefGoogle Scholar
[37]Muhly, P. S. and Tomforde, M.. Topological quivers. Internat. J. Math. 16 (2005), 693–755.CrossRefGoogle Scholar
[38]Okayasu, R.. Type III factors arising from Cuntz–Krieger algebras. Proc. Amer. Math. Soc. 131 (2002), 2145–2153.CrossRefGoogle Scholar
[39]Olsen, D. and Pedersen, G. K.. Some C *-dynamical systems with a single KMS state. Math. Scand. 42 (1978), 111–118.CrossRefGoogle Scholar
[40]Pimsner, M.. A class of C *-algebras generating both Cuntz–Krieger algebras and crossed product by . Free Probability Theory. American Mathematical Society, Providence, RI, 1997, pp. 189–212.Google Scholar
[41]Pinzari, C., Watatani, Y. and Yonetani, K.. KMS states, entropy and the variational principle in fullC *-dynamical systems. Comm. Math. Phys. 213 (2000), 331–379.CrossRefGoogle Scholar
[42]Renault, J.. Cuntz-like algebras. Operator Theoretical Methods (Timisoara, 1998). Theta Foundation, Bucharest, 2000, pp. 371–386.Google Scholar
[43]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading, MA, 1978.Google Scholar
[44]Ushiki, S.. Julia sets with polynomial symmetries. Proc. Int. Conf. on Dynamical Systems and Related Topics. World Scientific, Singapore, 1991.Google Scholar
[45]Zacharias, J.. Quasi-free automorphisms of Cuntz–Pimsner algebras. C*-algebras (Münsater, 1999). Springer, Berlin, 2000, pp. 262–272.CrossRefGoogle Scholar