Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-18T18:26:45.433Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometry of bifurcation currents for quadratic rational maps

Published online by Cambridge University Press:  14 March 2014

FRANÇOIS BERTELOOT  and
THOMAS GAUTHIER
FRANÇOIS BERTELOOT
Affiliation:
Université Paul Sabatier MIG, Institut de Mathématiques de Toulouse, 31062 Toulouse Cedex 9, France email berteloo@picard.ups-tlse.fr
THOMAS GAUTHIER
Affiliation:
Université de Picardie Jules Verne, LAMFA UMR-CNRS 7352, 80039 Amiens Cedex 1, France email thomas.gauthier@u-picardie.fr Stony Brook University, Institute for Mathematical Sciences, Stony Brook, NY 11794, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive $(1,1)$-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 5 , August 2015 , pp. 1369 - 1379
Copyright
© Cambridge University Press, 2014 

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

Berteloot, F.. Bifurcation currents in holomorphic families of rational maps. Pluripotential Theory (Lecture Notes in Mathematics, 2075). Springer, Berlin, 2013, pp. 1–93.Google Scholar
Branner, B.. The Mandelbrot set. Chaos and Fractals (Providence, RI, 1988) (Proceedings of Symposia in Applied Mathematics, 39). American Mathematical Society, Providence, RI, 1989, pp. 75–105.Google Scholar
Bassanelli, G. and Berteloot, F.. Bifurcation currents in holomorphic dynamics on ℙk. J. Reine Angew. Math. 608 (2007), 201–235.Google Scholar
Bassanelli, G. and Berteloot, F.. Distribution of polynomials with cycles of given mutiplier. Nagoya Math. J. 201 (2011), 23–43.CrossRefGoogle Scholar
Bassanelli, G. and Berteloot, F.. Lyapunov exponents, bifurcation currents and laminations in bifurcation loci. Math. Ann. 345(1) (2009), 1–23.CrossRefGoogle Scholar
Buff, X. and Epstein, A. L.. Bifurcation measure and postcritically finite rational maps. Complex Dynamics: Families and Friends. Ed. Schleicher, D.. A K Peters, Ltd., Wellesley, MA, 2009, pp. 491–512.CrossRefGoogle Scholar
Buff, X. and Gauthier, T.. Perturbations of flexible Lattès maps. Bull. Soc. Math. France to appear.Google Scholar
Bullett, S. and Sentenac, P.. Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Cambridge Philos. Soc. 115(3) (1994), 451–481.CrossRefGoogle Scholar
Demailly, J.-P.. Complex analytic and differential geometry, 2011. Freely accessible book (http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf).Google Scholar
DeMarco, L.. Dynamics of rational maps: a current on the bifurcation locus. Math. Res. Lett. 8(1–2) (2001), 57–66.CrossRefGoogle Scholar
DeMarco, L.. Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326(1) (2003), 43–73.CrossRefGoogle Scholar
DeMarco, L.. The moduli space of quadratic rational maps. J. Amer. Math. Soc. 20(2) (2007), 321–355.CrossRefGoogle Scholar
Dujardin, R.. Bifurcation currents and equidistribution on parameter space. Proceedings of the Conference ‘Frontiers in Complex Dynamics—Celebrating John Milnor’s 80th Birthday’, to appear. Preprint, 2011, math.DS/1111.3989.Google Scholar
Epstein, A. L.. Bounded hyperbolic components of quadratic rational maps. Ergod. Th. & Dynam. Sys. 20(3) (2000), 727–748.CrossRefGoogle Scholar
Goldberg, L. R. and Keen, L.. The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math. 101(2) (1990), 335–372.CrossRefGoogle Scholar
Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 5th edn. The Clarendon Press; Oxford University Press, New York, 1979.Google Scholar
Kiwi, J. and Rees, M.. Counting hyperbolic components. J. Lond. Math. Soc. to appear. Preprint, 2010. arXiv: math.DS/1003.6104v1.Google Scholar
Milnor, J.. Geometry and dynamics of quadratic rational maps. Experiment. Math. 2(1) (1993), 37–83 With an appendix by the author and Lei Tan.CrossRefGoogle Scholar
Mañé, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 193–217.CrossRefGoogle Scholar
Petersen, C. L.. No elliptic limits for quadratic maps. Ergod. Th. & Dynam. Sys. 19(1) (1999), 127–141.CrossRefGoogle Scholar