Thurston equivalence for rational maps with clusters (Q2842233)

The author considers bicritical rational maps of the Riemann sphere with two distinct super-attracting periodic orbits of the same period. He proves that some combinatorial data implies Thurston equivalence.NEWLINENEWLINEIf critical orbit Fatou components meet at a common boundary point, a cluster point is defined to be a point (in the Julia set) which is ``the endpoint of the angle 0 internal rays of at least one critical orbit Fatou component from each of the two critical cycles''. A cluster is the union of a cluster point and the Fatou components meeting it. The image of a cluster is a cluster, so one obtains cluster cycles. The relative orbits of the rays gives combinatorial data.NEWLINENEWLINEThe author proves the following results.NEWLINENEWLINETheorem A. Suppose that \(F\) and \(G\) are bicritical rational maps with fixed cluster cycles with the same combinatorial data. Then \(F\) and \(G\) are equivalent in the sense of Thurston.NEWLINENEWLINETheorem B. Suppose that two quadratic rational maps \(F\) and \(G\) have a period-two cluster cycle with rotation number \(p/n\) and critical displacement \(\delta\). Then \(F\) and \(G\) are equivalent in the sense of Thurston.NEWLINENEWLINERelated work includes [the author et al., J. Lond. Math. Soc., II. Ser. 87, No. 1, 87--110 (2013; Zbl 1333.37033); Ann. Fac. Sci. Toulouse, Math. (6) 21, No. 5, 907--934 (2012; Zbl 1343.37035)].