Mathematical Research Letters

Volume 8 (2001)

Number 1

Dynamics of rational maps: a current on the bifurcation locus

Pages: 57 – 66

DOI: http://dx.doi.org/10.4310/MRL.2001.v8.n1.a7

Author

Laura DeMarco (Harvard University)

Abstract

Let $f_\lambda:\P^1\to\P^1$ be a family of rational maps of degree $d >1$, parametrized holomorphically by $\lambda$ in a complex manifold $X$. We show that there exists a canonical closed, positive (1,1)-current $T$ on $X$ supported exactly on the bifurcation locus $B(f)\subset X$. If $X$ is a Stein manifold, then the stable regime $X-B(f)$ is also Stein. In particular, each stable component in the space $\Poly_d$ (or $\mbox{Rat}_d$) of all polynomials (or rational maps) of degree $d$ is a domain of holomorphy.

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