Dynamics of semiregular polynomial maps
Let \(f = (P_1,\ldots,P_k)\) be a proper polynomial map of \(\mathbb{C}^k\) and \(d_1 \geq \ldots \geq d_k\) be the degrees of the coordinates. We note \(\alpha\) the escape function rate on \(\mathbb{C}^k\), defined as \(\alpha(z) := \limsup {1 \over n} \log^+ \log^+ abs { f^n (z) }\). Observe that if \(f\) extends holomorphically to the projective space \(\mathbb{P}^k\), then \(\alpha\) takes its value in \(\{0,d\}\), where \(d\) is the common degree of the \(P_i\). In that situation, one may introduce the Green function, the Green current and the equilibrium measure of \(f\) [see \textit{N. Sibony} in: D. Cerveau et al., Dynamique et géométrie complexes. Paris: Société Mathématique de France. Panor. Synth. 8, 97--185 (1999; Zbl 1020.37026)]. The article deals with the more general case of ``regular maps''. The function \(\alpha\) then takes its value in \(\{0,d_1,\ldots,d_k\}\). It is also possible to build the equilibrium measure \(\mu\), by considering the partial Green functions defined on the closed subset \(\{ \alpha \leq d_i \}\). The proof relies on analysis of psh functions with respect to closed positive currents. The authors give also a lower estimate of the dimension of \(\mu\) in terms of the topological degree and the topological Lyapounov exponent of \(f\). Note that the preceding result remain valid for the larger class of ``semi-regular maps'', which are regular maps composed at the source with a projective polynomial map.
scientific article; zbMATH DE number 2103839
scientific article