Triviality of fibers for Misiurewicz parameters in the exponential family (Q2888641)

Local connectivity of the Mandelbrot set is perhaps the most celebrated open problem in complex dynamics. It is in fact a question concerning rigidity: whether certain combinatorial classes (the fibers of the projection from the Mandelbrot set to a well-known topological model) are trivial.NEWLINENEWLINEIn the exponential family, \(f_c:z\mapsto e^z+c\), a similar conjecture regarding triviality of fibers has been proposed by Schleicher; see [the reviewer and \textit{D. Schleicher}, in: Holomorphic dynamics and renormalization. A volume in honour of John Milnor's 75th birthday. Proceedings of the workshop on holomorphic dynamics, Toronto, Canada, 2006. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences. Fields Institute Communications 53, 177--196 (2008; Zbl 1154.37352)]. As for the Mandelbrot set, triviality of fibers in the exponential family would imply density of hyperbolicity in this parameter space.NEWLINENEWLINEIn the paper under review, the author proves triviality of fibers for Misiurewicz parameters; i.e., for those values \(c\) for which the omitted value \(c\) is eventually periodic under \(f_c\). (For the precise meaning of this statement, we refer to the article under review, or the paper by Schleicher and the reviewer cited above.)NEWLINENEWLINEThe proof uses a good understanding of the combinatorial structure of the exponential family (which can be thought of, combinatorially, as the limit of the families \(z^d+c\) of unicritical polynomials), to prove the following statement in the dynamical plane: If \(c'\) is close to \(c\), then the (dynamical) fiber of the periodic orbit obtained by analytic continuation of the postsingular periodic orbit of \(f_c\) is trivial. (The corresponding statement for polynomials would be that the Julia set is locally connected at this repelling periodic orbit.) This result is then transferred to the parameter plane.