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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)

##### MSC:
 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
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