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Dynamics of polynomial-like mappings. (Dynamique des applications d’allure polynomiale.) (French) Zbl 1033.37023
The authors study the dynamical properties of polynomial-like maps in several complex variables. They are proper holomorphic maps with nontrivial topological degree from \(U\) into \(V\), where \(U\) is a relatively compact set in a Stein manifold \(V\). Note that in dimension 1, such maps were studied by A. Douady and J. H. Hubbard [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 287–343 (1985; Zbl 0587.30028)].
In higher dimension, Dinh and Sibony prove that polynomial-like maps have mixing invariant measures of maximal entropy, whose support is in the set of points of a bounded orbit (the filled Julia set). If the plurisubharmonic functions are integrable with respect to such a measure \(m\), then \(m\) is exponentially mixing and its Lyapunov exponents are strictly positive. Moreover, the preimages of points (outside a union of an analytic subset of \(V\)) and the repulsive periodic points are equidistributed with respect to \(m\).
The proofs are based on functional analysis and pluripotential theory. Some arguments rely on the dynamics of real ramified coverings. The authors also observe that the “straightening theorem” of Douady-Hubbard in dimension 1 is no more true in higher dimension : they give an example of a polynomial-like map which is not topologically conjugated to a polynomial map with the same topological degree.

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37A25 Ergodicity, mixing, rates of mixing
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
32U05 Plurisubharmonic functions and generalizations
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