Perturbations of examples of Lattès and Hausdorff dimension of bifurcation place (Q1653064)

\textit{M. Shishikura} proved in [Ann. Math. (2) 147, No. 2, 225--267 (1998; Zbl 0922.58047)] that the Hausdorff dimension of the boundary of the Mandelbrot set $M$ is 2. A basic step in the proof is a {transfer principle} from the dynamical plane to the parameter plane, showing that if a parameter $c_0$ in the Mandelbrot set admits an invariant hyperbolic repeller of dimension $\delta$, then $M$ has dimension at least $\delta$ near $c_0$. Recently, in collaboration with \textit{C. Dupont} the authors provided in [Ann. Sci. Éc. Norm. Supér. (4) 51, No. 1, 215--262 (2018; Zbl 1454.32011)] a stability/bifurcation theory for the dynamics of rational endomorphisms of projective spaces. It is natural to try to estimate the Hausdorff dimension of the bifurcation locus. It is known that for certain families the bifurcation locus can have a non-empty interior (see [\textit{F. Bianchi} and \textit{J. Taflin}, Proc. Am. Math. Soc. 145, No. 10, 4337--4343 (2017; Zbl 1375.32037); \textit{R. Dujardin}, J. Éc. Polytech., Math. 4, 813--843 (2017; Zbl 1406.37041)]). In the paper under review, the authors prove a higher-dimensional version of Shishikura's transfer principle. The lack of conformality makes it more delicate than the one-dimensional case. It is easy to construct hyperbolic repellers of a dimension arbitrarily close to $2k$ for Lattès maps of $\mathbb{P}^k(\mathbb{C})$ belonging to the bifurcation locus. Applying the transfer principle, the authors are able to prove that the bifurcation locus has maximal Hausdorff dimension at such parameters.