zbMATH — the first resource for mathematics

On the geometry of bifurcation currents for quadratic rational maps. (English) Zbl 1339.37035
This paper reviews some important results concerning the bifurcation in the dynamics of functions, and then studies the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. The authors obtain an appropriate counterpart of J. Milnor’s [Exp. Math. 2, No. 1, 37–83 (1993; Zbl 0922.58062)] and A. L. Epstein’s [Ergodic Theory Dyn. Syst. 20, No. 3, 727–748 (2000; Zbl 0963.37041)] results in the the setting of currents. The extension of some results are given for some closed, positive \((1,1)\)-current on a two-dimensional complex projective space. The computation of the Lelong numbers and the self-intersection of the extended current are also described. It is proved that the mass of the measure coincides with the Lelong number of the bifurcation current at the point. Finally, some important differences between the moduli space of quadratic rational maps and the moduli space of cubic polynomials are shown.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37P45 Families and moduli spaces in arithmetic and non-Archimedean dynamical systems
Full Text: DOI arXiv
[1] DOI: 10.1007/s00208-008-0325-1 · Zbl 1179.37067 · doi:10.1007/s00208-008-0325-1
[2] DOI: 10.1201/b10617-16 · doi:10.1201/b10617-16
[3] Bassanelli, J. Reine Angew. Math. 608 pp 201– (2007)
[4] Branner, Chaos and Fractals (Providence, RI, 1988) pp 75– (1989)
[5] DOI: 10.1007/978-3-642-36421-1_1 · Zbl 1280.37039 · doi:10.1007/978-3-642-36421-1_1
[6] Mañé, Ann. Sci. Éc. Norm. Supér. (4) 16 pp 193– (1983)
[7] DOI: 10.1017/S0143385799120996 · Zbl 0921.30019 · doi:10.1017/S0143385799120996
[8] DOI: 10.1080/10586458.1993.10504267 · Zbl 0922.58062 · doi:10.1080/10586458.1993.10504267
[9] Hardy, An Introduction to the Theory of Numbers, 5th edn (1979) · Zbl 0423.10001
[10] DOI: 10.1007/BF01231505 · Zbl 0715.58018 · doi:10.1007/BF01231505
[11] DOI: 10.1017/S0143385700000390 · Zbl 0963.37041 · doi:10.1017/S0143385700000390
[12] DOI: 10.1090/S0894-0347-06-00527-3 · Zbl 1158.37020 · doi:10.1090/S0894-0347-06-00527-3
[13] DOI: 10.1007/s00208-002-0404-7 · Zbl 1032.37029 · doi:10.1007/s00208-002-0404-7
[14] DOI: 10.4310/MRL.2001.v8.n1.a7 · Zbl 0991.37030 · doi:10.4310/MRL.2001.v8.n1.a7
[15] DOI: 10.1017/S0305004100072236 · Zbl 0823.58012 · doi:10.1017/S0305004100072236
[16] DOI: 10.1215/00277630-2010-016 · Zbl 1267.37049 · doi:10.1215/00277630-2010-016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.