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Equidistribution towards the bifurcation current. I: Multipliers and degree \(d\) polynomials. (English) Zbl 1380.37098
Let \(\mathcal{P}_d\) be the moduli space (under affine conjugacy) of degree-\(d\) complex polynomials with \(d-1\) marked critical points. The bifurcation locus is the closure in \(\mathcal{P}_d\) of the set of discontinuity of the map associating to \(P\in\mathcal{P}_d\) its Julia set, endowed with the bifurcation current \(T_{\mathrm{bif}}\) (see, e.g., [L. DeMarco, Math. Res. Lett. 8, No. 1–2, 57–66 (2001; Zbl 0991.37030)]) defined as \(dd^cL\), where \(L\) is the Lyapunov exponent of \(P\).
The main result of this paper is the fact that \(T_{\mathrm{bif}}\) can be approximated (in the sense of currents) by the hypersurfaces of polynomials having periodic points of fixed multiplier. More precisely, if \(\mathrm{Per}_n(w)\) denotes the set (known to be an algebraic hypersurface) of \(p\in\mathcal{P}_d\) having a cycle of exact period \(n\) and multiplier \(w\in\mathbb{C}\) then \(d^{-n}[\mathrm{Per}_n(w)]\) converges in the weak sense of currents to \(T_{\mathrm{bif}}\) as \(n\to+\infty\). This generalises previous results by G. Bassanelli and F. Berteloot [Math. Ann. 345, No. 1, 1–23 (2009; Zbl 1179.37067); Nagoya Math. J. 201, 23–43 (2011; Zbl 1267.37049)] valid for \(|w|\leq 1\), and by X. Buff and the author [Proc. Am. Math. Soc. 143, No. 7, 3011–3017 (2015; Zbl 1334.37038)] valid for \(d=2\).
An important step in the proof is a new comparison theorem for plurisubharmonic functions. Let \(X\) be a complex manifold of dimension \(k\geq 1\) admitting a smooth plurisubharmonic function \(w\) such that \((dd^cw)^k\) is a non-degenerate volume form outside a strict analytic subset. Let \(\Omega\subset X\) be a \(C^1\) domain, and \(u\), \(v\) plurisubharmonic functions defined on \(\Omega\), with \(v\) continuous and such that \((dd^cv)^k\) has finite mass. Assume that there exists a compact subset \(K\subset\subset\Omega\) such that \(\mathrm{supp}((dd^c v)^k)\subset\partial K\) and \((dd^cv)^k(\partial U)=0\) for any connected component of the interior of \(K\). Then the author proves that if \(u\leq v\) on \(\Omega\) and \(u\equiv v\) on \(\Omega\setminus K\) then \(u\equiv v\) on \(\Omega\).
Reviewer: Marco Abate (Pisa)

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32U40 Currents
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
Full Text: DOI
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