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Distribution of postcritically finite polynomials. III: Combinatorial continuity. (English) Zbl 1432.37074
A combinatorial approach based on external rays is used to study the distribution of postcritically finite (PCF) parameters in the moduli space of polynomials. Arithmetic methods were used by C. Favre and T. Gauthier [Isr. J. Math. 209, Part 1, 235–292 (2015; Zbl 1352.37202)] to prove a result showing the equidistribution of PCF parameters with respect to the bifurcation measure, with an exponential speed of convergence. The current work sets conditions on the combinatorics of angles with given period and preperiod landing at critical points, instead of requiring conditions on the parameter itself, to obtain the equidistribution of PCF Misiurewicz parameters with respect to the bifurcation measure.
The result may be viewed as supplying a measurable version of work of R. Dujardin and C. Favre [Am. J. Math. 130, No. 4, 979–1032 (2008; Zbl 1246.37071)] using results of J. Kiwi [Proc. Lond. Math. Soc. (3) 91, No. 1, 215–248 (2005; Zbl 1077.37038)] on the combinatorial space and the landing of external rays. In [J. Mod. Dyn. 11, 57–98 (2017; Zbl 1419.37040)], the authors earlier concerned themselves with speed of convergence of the equidistribution of PCF parameters with respect to the bifurcation measure.

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
30D40 Cluster sets, prime ends, boundary behavior
37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets
Full Text: DOI
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