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The bifurcation measure has maximal entropy. (English) Zbl 1437.37061
By the analogy between the dynamics of an endomorphism of \(\mathbb{P}^k\) and bifurcation in a holomorphic family of rational maps, the main goal of the paper is to show that the bifurcation measure has maximal entropy (Theorem B).
For this purpose, the authors first introduce the notions of bifurcation entropy and metric bifurcation entropy, based on the concept of a \((d_n,\epsilon)\)-separated set, where \(d_n\) is the so-called \(n\)-bifurcation distance on the parameter space. One of the key ingredients is a generalization of Yomdin’s bound of the volume of the image of a dynamical ball.
The authors give an alternative proof of the computation of the entropy of the Green measure of Hénon maps, and define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of \(\mathbb{P}^k\).
37F80 Higher-dimensional holomorphic and meromorphic dynamics
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets
32U40 Currents
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
Full Text: DOI
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