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Degree complexity of a family of birational maps. (English) Zbl 1153.37387

Summary: We compute the degree complexity of a family of birational mappings of the plane with high order singularities.

MSC:

37F99 Dynamical systems over complex numbers
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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[1] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: From Yang-Baxter equations to dynamical zeta functions for birational transformations. Statistical physics on the eve of the 21st century. Ser. Adv. Statist. Mech. 14, 436–490 (1999)
[2] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Rational dynamical zeta functions for birational transformations. Phys. A 264, 264–293, chao-dyn/9807014 (1999) · doi:10.1016/S0378-4371(98)00452-X
[3] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Topological entropy and complexity for discrete dynamical systems. Phys. Lett. A 262, 44–49, chao-dyn/9806026 (1999) · Zbl 0936.37005 · doi:10.1016/S0375-9601(99)00662-3
[4] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Growth complexity spectrum of some discrete dynamical systems. Phys. D 130(1–2), 27–42 (1999) · Zbl 0940.37017 · doi:10.1016/S0167-2789(99)00014-7
[5] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Real topological entropy versus metric entropy for birational measure-preserving transformations. Phys. D 144(3–4), 387–433 (2000) · Zbl 0978.37036 · doi:10.1016/S0167-2789(00)00079-8
[6] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Real Arnold complexity versus real topological entropy for birational transformations. J. Phys. A 33(8), 1465–1501 (2000) · Zbl 0983.37021 · doi:10.1088/0305-4470/33/8/301
[7] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Topological entropy and Arnold complexity for two-dimensional mappings. Phys. Lett. A 262(1), 44–49 (1999) · Zbl 0936.37005 · doi:10.1016/S0375-9601(99)00662-3
[8] Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Elliptic curves from finite order recursions or non-involutive permutations for discrete dynamical systems and lattice statistical mechanics. Eur. Phys. J. B, 647–661 (1998)
[9] Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Functional relations in lattice statistical mechanics, enumerative combinatorics and discrete dynamical systems. Ann. Comb. 3, 131–158 (1999) · Zbl 1055.82505 · doi:10.1007/BF01608780
[10] Bedford, E., Diller, J.: Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift. Amer. Math. J. 127(3), 595–646 (2005) · Zbl 1083.37038 · doi:10.1353/ajm.2005.0015
[11] Bedford, E., Diller, J.: Dynamics of a two parameter family of plane birational maps: maximal entropy. J. Geom. Anal. 16(3), 409–430 (2006) · Zbl 1116.37031
[12] Bedford, E., Diller, J.: Real dynamics of a family of plane birational maps: trapping regions and entropy zero. arXiv.math/0609113 (2006) · Zbl 1116.37031
[13] Bedford, E., Kim, K.H.: Dynamics of rational surface automorphisms: linear fractional recurrences. arXiv:math/0611297 (2007) · Zbl 1185.37128
[14] Bellon, M.P., Maillard, J.-M., Viallet, C.-M.: Quasi-integrability of the sixteen vertex model. Phys. Lett. B 281, 315–319 (1992) · Zbl 0797.17025 · doi:10.1016/0370-2693(92)91147-2
[15] Bellon, M.P., Maillard, J.-M., Viallet, C.-M.: Dynamical systems from quantum integrability. In: Maillard, J.-M. (ed.) Proceedings of the Conference ”Yang-Baxter Equations in Paris”, pp. 95–124 World Scientific, Singapore (1993) (also published as a supplement of Int. J. Mod. Phys.) · Zbl 0820.17039
[16] Bellon, M., Viallet, C.: Algebraic entropy. Comm. Math. Phys. 204, 425–437 (1999) · Zbl 0987.37007 · doi:10.1007/s002200050652
[17] Bose, R.C., Mesner, D.M.: On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 10, 21–38 (1959) · Zbl 0089.15002 · doi:10.1214/aoms/1177706356
[18] Bouamra, M., Boukraa, S., Hassani, S., Maillard, J.-M.: Post-critical set and preserved meromorphic two-forms. J. Phys. A 38, 7957–7988, nlin.CD/0505024 v1 (2005) · Zbl 1087.37040 · doi:10.1088/0305-4470/38/37/003
[19] Boukraa, S., Hassani, S., Maillard, J.-M.: Product of involutions and fixed points. Alg. Rev. Nucl. Sci. 2, 1–16 (1998)
[20] Boukraa, S., Maillard, J.-M.: Factorization properties of birational mappings. Phys. A 220, 403–470 (1995) · doi:10.1016/0378-4371(95)00220-2
[21] Boukraa, S., Maillard, J.-M., Rollet, G.: Almost integrable mappings. Int. J. Mod. Phys. B8, 137–174 (1994) · Zbl 0804.58045
[22] Boukraa, S., Maillard, J.-M., Rollet, G.: Integrable mappings and polynomial growth. Phys. A 209, 162–222 (1994) · Zbl 0804.58045 · doi:10.1016/0378-4371(94)90055-8
[23] Boukraa, S., Maillard, J.-M., Rollet, G.: Determinental identities on integrable mappings. Int. J. Mod. Phys. B8, 2157–2201 (1994) · Zbl 1264.82044
[24] Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123, 1135–1169 (2001) · Zbl 1112.37308 · doi:10.1353/ajm.2001.0038
[25] Fornæss, J.-E., Sibony, N.: Complex dynamics in higher dimension, II, Modern methods in complex analysis. Ann. Math. Stud. 137, 135–182 (1995) · Zbl 0847.58059
[26] Gizatullin, M.: Rational G-surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 44, 110–144 (1980) · Zbl 0428.14022
[27] Hansel, D., Maillard, J.-M.: Symmetries of models with genus >1. Phys. Lett. A 133, 11–15 (1988) · doi:10.1016/0375-9601(88)90726-8
[28] Jaeger, F.: Towards a classification of spin models in terms of association schemes, Progress in algebraic combinatorics (Fukuoka 19993). Math. Soc. Japan 24, 197–225 (1996) [10, 21–38 (1959)]
[29] Jaekel, M.T., Maillard, J.-M.: Symmetry relations in exactly soluble models. J. Phys. A 15, 1309–1325 (1982) · doi:10.1088/0305-4470/15/4/031
[30] Jaekel, M.T., Maillard, J.-M.: Inverse functional relations on the Potts model. J. Phys. A 15, 2241–2257 (1982) · doi:10.1088/0305-4470/15/7/034
[31] Jaekel, M.T., Maillard, J.-M.: Inversion functional relations for lattice models. J. Phys. A 16, 1975–1992 (1983) · doi:10.1088/0305-4470/16/9/022
[32] Maillard, J.-M.: Automorphisms of algebraic varieties and Yang-Baxter equations. J. Math. Phys. 27, 2776–2781 (1986) · Zbl 0609.14028 · doi:10.1063/1.527303
[33] Meyer, H., Anglès d’Auriac, J.-C., Maillard, J.-M., Rollet, G.: Phase diagram of a six-state chiral Potts model. Phys. A 208, 223–236 (1994) · doi:10.1016/0378-4371(94)90056-6
[34] Quispel, G.R.W., Roberts, J.A.G.: Reversible mappings of the plane. Phys. Lett. A 132, 161–163 (1988) · doi:10.1016/0375-9601(88)90274-5
[35] Quispel, G.R.W., Roberts, J.A.G.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216, 63–177 (1992) · doi:10.1016/0370-1573(92)90163-T
[36] Syozi, I.: Transformation of Ising models. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 1, pp. 269–329. Academic, London (1972)
[37] Takenawa, T.: A geometric approach to singularity confinement and algebraic entropy. J. Phys. A 34, L95–L102 (2001) · Zbl 0988.37011 · doi:10.1088/0305-4470/34/10/103
[38] Takenawa, T.: Discrete dynamical systems associated with root systems of indefinite type. Comm. Math. Phys. 224, 657–681 (2001) · Zbl 1038.37507 · doi:10.1007/s002200100568
[39] Takenawa, T.: Algebraic entropy and the space of initial values for discrete dynamical systems. J. Phys. A 34, 10533–10545 (2001) · Zbl 0999.37028 · doi:10.1088/0305-4470/34/48/317
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