An ergodic study of Painlevé VI

Mathematische Annalen

Volumn 338, Issue 2, 2007, Pages 295-345

  Iwasaki, Katsunori ^a   Uehara, Takato ^a  

^a KYUSHU UNIVERSITY   (Japan)

Author keywords

Chaos; Entropy; Ergodic theory; Invariant measure; Painlev VI

Indexed keywords

EID: 34248224932     PISSN: 00255831     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00208-006-0077-8     Document Type: Article

References (40)

1. 4}. Int. Math. Res. Notices 1997(19), 983-999 (1997)
2. Bedford, E., Diller, J.: Energy and invariant measures for birational surface maps. Duke Math. J. 128(2), 331-368 (2005)
3. 2. Invent. Math. 114(2), 277-288 (1993)
4. Birman, J.S.: Braids, Links, and Mapping Class Groups. Ann. Math. Stud., Princeton University Press, Princeton (1974)
5. Boalch, P.: From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. 90(3), 167-208 (2005)
6. Boalch, P.: The fifty-two icosahedral solutions to Painlevé VI. preprint (arXiv:math. AG/0406281)
7. Cantat, S.: Dinamique des automorphismes des surfaces K3. Acta Math. 187(1), 1-57 (2001)
8. Cantat, S.: Sur la dynamique du groupe d'automorphismes des surfaces K3. Transform. Groups 6(3), 201-214 (2001)
9. Cantat, S.: Dynamique diophantienne. Images des Mathématiques, 41-46 (2004)
10. Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. Perseus Books, Reading (1989)
11. Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123(6), 1135-1169 (2001)
12. Dinh, T.-C., Sibony, N.: Une borne supérieure pour l'entropie topologique d'une application rationnelle. Ann. Math. 161(2) (3), 1637-1644 (2005)
13. Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141(1), 55-147 (2000)
14. Dujardin, R.: Laminar currents and birational dynamics. Duke Math. J. 131(2), 219-247 (2006)
15. 2. Ann. Inst. Fourier 48(4), 999-1023 (1998)
16. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)
17. Goldman, W.M.: Ergodic theory on moduli spaces. Ann. Math. 146 (2), 475-507 (1997)
18. Hitchin, N.: Poncelet polygons and the Painlevé equations. In: Geometry and Analysis, Bombay, 1992, 151-185. Tata Inst. Fund. Res., Bombay (1995)
19. Hitchin, N.: A lecture on the octahedron. Bull. London Math. Soc. 35(5), 577-600 (2003)
20. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
21. Inaba, M., Iwasaki, K., Saito, M.-H.: Dynamics of the sixth Painlevé equation. In: Loday, M., Delabaere, E. (eds.) Théorie asymptotique et équations de Painlevé, Angers, juin 2004, Séminaires et Congrès 14. Soc. Math. France (in press) (arXiv:math.AG/0501007)
22. Inaba, M., Iwasaki, K., Saito, M.-H.: Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. Part I. In: Publ. Res. Inst. Math. Sci. 42(4), 987-1089 (2006)
23. Inaba, M., Iwasaki, K., Saito, M.-H.: Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. Part II. Adv. Stud. Pure Math. 45, 387-432 (2006)
24. Iwasaki, K.: Moduli and deformation for Fuchsian projective connections on a Riemann surface. I Fac. Sci. Univ. Tokyo Sect. IA Math. 38(3), 431-531 (1991)
25. Iwasaki, K.: Fuchsian moduli on a Riemann surface-its Poisson structure and Poincaré-Lefschetz duality. Pacific J. Math. 155, 319-340 (1992)
26. Iwasaki, K.: A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation. Proc. Japan Acad. Ser. A 78, 131-135 (2002)
27. Iwasaki, K.: An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation. Comm. Math. Phys. 242(1-2), 185-219 (2003)
28. Iwasaki, K.: Bounded trajectories of the sixth Painlevé equation (tentative title) (in preparation)
29. Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci. 18(3), 1137-1161 (1982)
30. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)
31. Kitaev, A.V.: Grothendieck's dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations. Algebra i Analiz 17(1), 224-275 (2005)
32. Mazzocco, M.: Rational solutions of the Painlevé VI equation. J. Phys. A: Math. Gen. 34, 2281-2294 (2001)
33. McMullen, C.T.: Dynamics on K3 surfaces: Salem numbers and Siegel disks. J. Reine Angew. Math. 545, 201-233 (2002)
34. Mumford, D.: Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34. Springer, Berlin (1965)
35. Okamoto, K.: Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Espaces Des Conditions Initiales. Jpn. J. Math. 5, 1-79 (1979)
36. Ornstein, D.S., Weiss, B.: Statistical properties of chaotic systems. Bull. Amer. Math. Soc. (N.S.) 24, 11-116 (1991)
37. Saito, M.-H., Takebe, T., Terajima, H.: Deformation of Okamoto-Painlevé pairs and Painlevé equations. J. Algebr. Geom. 11(2), 311-362 (2002)
38. Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys. 220, 165-229 (2001)
39. 2. Ergodic Theory Dynam. Systems 10(4), 823-827 (1990)
40. Walters, P: An introduction to ergodic theory, GTM 79. Springer, New York (1982)