Finiteness for degenerate polynomials

Details Finiteness for degenerate polynomials Finiteness for degenerate polynomials

2006-08-31

arxiv.org/abs/math/0608800v1

Mathematics

Dynamical Systems

15 pages, for the Proceedings of the Holomorphic Dynamics Workshop, in celebration of J. Milnor's 75th birthday

Scientific paper

Let $\MP_d$ denote the space of polynomials $f: \C \to \C$ of degree $d\geq 2$, modulo conjugation by $\Aut(\C)$. Using properties of polynomial trees (as introduced in [DM, math.DS/0608759]), we show that if $f_n$ is a divergent sequence of polynomials in $\MP_d$, then any subsequential limit of the measures of maximal entropy $m(f_n)$ will have finite support. With similar techniques, we observe that the iteration maps $\{\MPbar_d \dashrightarrow \MPbar_{d^n}: n\geq 1\}$ between GIT-compactifications can be resolved simultaneously with only finitely many blow-ups of $\MPbar_d$.

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