Dynamics of polynomial-like mappings
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 \textit{A. Douady} and \textit{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.
mixing invariant measures of maximal entropy
scientific article; zbMATH DE number 1985982
scientific article