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Random walks, Kleinian groups, and bifurcation currents

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Abstract

Let (ρ λ ) λ∈Λ be a holomorphic family of representations of a finitely generated group G into PSL(2,ℂ), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λ describing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on ℙ1. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μ on G.

We show that under natural assumptions on μ, the support of the bifurcation current coincides with the bifurcation locus of the family. We also prove that the bifurcation current describes the asymptotic distribution of several codimension 1 phenomena in parameter space, like accidental parabolics or new relations, or accidental collisions between fixed points.

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Authors and Affiliations

  1. CNRS, Département de Mathématique d’Orsay, Bâtiment 425, Université de Paris Sud, 91405, Orsay cedex, France

    Bertrand Deroin

  2. CMLS, École Polytechnique, 91128, Palaiseau, France

    Romain Dujardin

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  1. Bertrand Deroin
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  2. Romain Dujardin
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Correspondence to Romain Dujardin.

Additional information

B.D.’s research was partially supported by ANR-08-JCJC-0130-01, ANR-09-BLAN-0116. R.D.’s research was partially supported by ANR project BERKO and by ECOS project C07E01.

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Deroin, B., Dujardin, R. Random walks, Kleinian groups, and bifurcation currents. Invent. math. 190, 57–118 (2012). https://doi.org/10.1007/s00222-012-0376-5

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