Julia sets on \(\mathbb{R}\mathbb{P}^2\) and dianalytic dynamics (Q2922923)

A map \(f:\mathbb{RP}^2\to\mathbb{RP}^2\) is \textit{dianalytic} if it lifts to a rational map of the Riemann sphere \(\mathbb{C}_\infty\) by the natural double cover \(\pi:\mathbb{C}_\infty\rightarrow\mathbb{RP}^2\). In the article under review, the authors study the dynamics of dianalytic mappings of the real projective plane \(\mathbb{RP}^2\). They first establish connections between dianalytic mappings and quasi-real rational maps, i.e., rational maps which are conformally conjugate to their complex conjugate. In particular, the authors classify bicritical quasi-real rational maps of degree \(n\). Namely, they prove that any bicritical dianalytic mapping lifts to a quasi-real rational map which is conformally conjugate to a rational map of the form \(f(z)=\frac{az^n+b}{-\bar{b}z^n+\bar{a}}\); see Theorem 2.6.NEWLINENEWLINEIn a second time, they use Theorem 2.6 to define the moduli space of dianalytic mappings in Section 3 and to describe precisely the moduli space of cubic dianalytic bicritical mappings in Section 5. Section 4 is devoted to studying the Julia set of dianalytic mappings. In particular, the authors describe its symmetries and they provide a wide class of bicritical dianalytic mappings of odd degree having a quasi-circle Julia set.