GIT Stability of Henon Maps

In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $operatorname{Rat}_d^N$ of degree $d$ rational maps $mathbb{P}^Ntomathbb{P}^N$ modulo the conjugation action of $operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $Nge3$ or $dge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $operatorname{Rat}_2^2$.