×

zbMATH — the first resource for mathematics

Bifurcation currents in holomorphic dynamics on \(\mathbb P^k\). (English) Zbl 1136.37025
Authors’ abstract: We use pluri-potential theory to study the bifurcations of holomorphic families \(\{f_\lambda\}_{\lambda\in X}\) of rational maps on \(\mathbb{P}^1\) or endomorphisms of \(\mathbb{P}^k\). To this purpose we establish some formulas for \(L(f_\lambda)\) and \(dd^cL(f_\lambda)\) where \(L(f_\lambda)\) is the sum of the Lyapunov exponents of \(f_\lambda\) with respect to the maximal entropy measure. We show that the bifurcation current \(dd^cL(f_\lambda)\) both detects the instability of repulsive cycles and the interaction between critical and Julia sets. For families of rational maps of degree \(d\), we introduce a bifurcation measure defined by \((dd^cL(f_\lambda))^{2d-2}\) and study its first properties. In particular, we show that the support of this measure is contained in the closure of the set of rational maps having \(2d-2\) distinct Cremer-cycles. This approach yields to a purely potential-theoretic proof of the Mané-Sad-Sullivan theorem and, moreover, allows us to extend it.
Reviewer: Pei-Chu Hu (Jinan)

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bedford E., J. Geom. Anal. 8 pp 3– (1998)
[2] DOI: 10.1090/S0002-9904-1976-13977-8 · Zbl 0322.31008 · doi:10.1090/S0002-9904-1976-13977-8
[3] DOI: 10.1007/BF02392572 · Zbl 1144.37436 · doi:10.1007/BF02392572
[4] Briend J.-Y., Publ. Math. Inst. Hautes E’t. Sci. 93 pp 145– (2001)
[5] DOI: 10.1007/BF02591353 · Zbl 0127.03401 · doi:10.1007/BF02591353
[6] Chen X., J. Di\currency. Geom. 56 pp 2– (2000)
[7] Demailly J. P., S.) pp 19– (1985)
[8] Demailly J. P., J. Alg. Geom. 1 pp 361– (1992)
[9] DeMarco L., Math. Res. Lett. 8 pp 1– (2001)
[10] DOI: 10.1007/s00208-002-0404-7 · Zbl 1032.37029 · doi:10.1007/s00208-002-0404-7
[11] Dinh T. C., J. Math. Pures Appl. (IX) 82 pp 4– (2003)
[12] Jonsson M., Erg. Th. Dyn. Syst. 18 pp 3– (1998)
[13] Ledrappier F., C. R. Acad. Sci. Paris Sér. I Math. 299 pp 1– (1984)
[14] Maé R., Ann. Sci. Ec. Norm. Supér. (IV) 16 pp 193– (1983)
[15] DOI: 10.1007/BFb0083068 · doi:10.1007/BFb0083068
[16] McMullen C. T., Ann. Math. Stud. pp 135– (1995)
[17] Milnor J. W., Experiment. Math. 2 pp 1– (1993)
[18] DOI: 10.1007/BF01388554 · Zbl 0569.58024 · doi:10.1007/BF01388554
[19] DOI: 10.1007/BF02584795 · Zbl 0432.58013 · doi:10.1007/BF02584795
[20] DOI: 10.2307/2374768 · Zbl 0790.32017 · doi:10.2307/2374768
[21] Shishikura M., Ann. Sci. Ec. Norm. Sup. (IV) 20 pp 1– (1987)
[22] Sibony N., Panor. Synth. 8 pp 97– (1999)
[23] DOI: 10.1215/S0012-7094-98-09404-2 · Zbl 0966.14031 · doi:10.1215/S0012-7094-98-09404-2
[24] DOI: 10.1007/BF01389965 · Zbl 0289.32003 · doi:10.1007/BF01389965
[25] DOI: 10.1007/BF01389217 · Zbl 0488.58002 · doi:10.1007/BF01389217
[26] DOI: 10.1007/BFb0072757 · doi:10.1007/BFb0072757
[27] DOI: 10.1007/BF01234434 · Zbl 0820.58038 · doi:10.1007/BF01234434
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.