Distribution of rational maps with a preperiodic critical point

By Romain Dujardin, C. Favre. 2006

130 citations recorded by the source corpus.

Let $\{f_\lambda\}$ be a family of rational maps of a fixed degree, with a marked critical point $c(\lambda)$. Under a natural assumption, we first prove that the hypersurfaces of parameters for which $c(\lambda)$ is periodic converge as a sequence of positive closed $(1,1)$ currents to the bifurcation current attached to $c$ and defined by DeMarco. We then turn our attention to the parameter space of polynomials of a fixed degree $d$. By intersecting the $d-1$ currents attached to each critical point of a polynomial, Bassaneli and Berteloot obtained a positive measure $\mu_{\rm bif}$ of finite mass which is supported on the connectedness locus. They showed that its support is included in the closure of the set of parameters admitting $d-1$ neutral cycles. We show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials (i.e., of Misiurewicz points).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5c9bb2a3cd2c78a059a2865eb6b4e9841be05e6b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Bifurcations, intersections, and heights (2014, background): “5 in [Dujardin and Favre 2008], we will need the following statement for our proof of Theorem 1.”
Bifurcation Currents in Holomorphic Families of Rational Maps (2012, methodology): “We refer to the paper [29], Proposition 6.”
SPECIAL CURVES AND POSTCRITICALLY FINITE POLYNOMIALS (2012, background): “hat case, the support of acoincides with the activity locus of the critical point, the set of parameters where the critical point is \passing through" the Julia set of f t [MSS]. See [De1] and [DF] for background on bifurcation currents. Similarly for any marked point, the support of the measu”
Preperiodic points and unlikely intersections (2009, background): “It follows from the arguments in [DF08] (specifically the proof of their Theorem 1) that the solutions c ∈ C to equations f c (a) = f k(n) c (a), weighted by multiplicity, will equidistribute to the equilibrium measure μMa on ∂Ma as n → ∞ for any sequence 0 ≤ k(n) < n.”