Dynamics of the projectively natural polygon iteration on projective polygon space

Let $n \geq 5$ and let $H_{\lambda}$ be the polygon iteration on the projective moduli space $\mathfrak{P}_n = (\mathbb{R}P^2)^n/PGL(3,\mathbb{R})$ of projective equivalence classes of generic $n$-gons. Let $P_{reg}$ denote the class of the regular polygon. For $m$-regular polygons, let $P_{m-reg}$ denote the class whose vertices are $\exp(2jm\pi i/n) \in \mathbb{C} \subset \mathbb{R}P^2$ for $j=0,\ldots,n-1$. The dynamics conjecture. If $\lambda \in (0,\infty)$, then for almost all $P \in \mathfrak{P}_n$, $\lim_{k \to \infty} H_{\lambda}^k(P)=P_{reg}. $ If $\lambda \in (-\infty,0)$, then: (i) if $n=3m\pm1$, for almost all $P \in \mathfrak{P}_n$, $\lim_{k \to \infty} H_{\lambda}^k(P)=P_{m-reg}; $ (ii) if $n=3m$, for almost all $P \in \mathfrak{P}_n$, the iterates $H_{\lambda}^k(P)$ escape all compact subsets of the set of generic $n$-gons. The conjecture predicts the limiting regular or star-regular behavior of the projectively natural iteration for almost every projective polygon. The stated convergence and degeneration behavior is suggested by computer simulations, while the supplied text gives no resolution of the conjecture.