Finiteness for degenerate polynomials
Abstract
Description
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 $\MPbar_d$.
15 pages, for the Proceedings of the Holomorphic Dynamics Workshop, in celebration of J. Milnor's 75th birthday
15 pages, for the Proceedings of the Holomorphic Dynamics Workshop, in celebration of J. Milnor's 75th birthday