Iteration at the boundary of the space of rational maps
Abstract
Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$, and $\mu_f$ the measure of maximal entropy for $f\in Rat_d$. The map of measures $f\mapsto\mu_f$ is known to be continuous on $Rat_d$, and it is shown here to extend continuously to the boundary of $Rat_d$ in $\bar{Rat}_d \simeq {\bf P}^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\mapsto f^n$ for every $n\geq 2$. The limiting measures are given explicitly, away from $I(d)$. The degenerations of rational maps are also described in terms of metrics of non-negative curvature on the Riemann sphere: the limits are polyhedral.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- March 2004
- DOI:
- 10.48550/arXiv.math/0403078
- arXiv:
- arXiv:math/0403078
- Bibcode:
- 2004math......3078D
- Keywords:
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- Dynamical Systems;
- Complex Variables
- E-Print:
- 25 pages