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Perturbations of examples of Lattès and Hausdorff dimension of bifurcation place. (Perturbations d’exemples de Lattès et dimension de Hausdorff du lieu de bifurcation.) (French. English summary) Zbl 1417.37171
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 C. Dupont the authors provided in [Ann. Sci. Éc. Norm. Supér. (4) 51, No. 1, 215–262 (2018; Zbl 06873715)] 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 [F. Bianchi and J. Taflin, Proc. Am. Math. Soc. 145, No. 10, 4337–4343 (2017; Zbl 1375.32037); 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.

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
Full Text: DOI
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