Valiron and Abel equations for holomorphic self-maps of the polydisc (Q2807997)

The paper under review deals with holomorphic self-maps \(f:\Delta^N\to\Delta^N\) of the polydisc. Using the divergence rate, defined as the limit NEWLINE\[NEWLINEc(f):=\lim_{m\to\infty}\frac{k_{\Delta^N}(f^m(z),z)}{m},NEWLINE\]NEWLINE where \(k_{\Delta^N}\) stands for the Kobayashi pseudo-distance, the authors introduce a notion of hyperbolicity and parabolicity: \(f\) is called parabolic if it has no fixed point and \(c(f)=0\), and is called hyperbolic if \(c(f)>0\) (which implies that \(f\) has no fixed point). These notions generalize the corresponding ones for the case \(N=1\), which are based on the dilation of \(f\) at the Denjoy-Wolff point.NEWLINENEWLINEThe main results of the paper are generalizations of the classical one-variable results which provide models for the dynamics of \(f\) in the hyperbolic and parabolic cases, using the term \(\lambda_f:=e^{-c(f)}\) as substitute of the dilation of \(f\) at the Denjoy-Wolff point. When \(f\) is hyperbolic, the authors prove that there exists a holomorphic function \(\Theta:\Delta^N\to\mathbb{H}\) that solves the Valiron equation NEWLINE\[NEWLINE\Theta(f(z))=\frac{1}{\lambda_f}\Theta(z)NEWLINE\]NEWLINE for all \(z\) and satisfies \(\bigcup_{n\geq0}\lambda_f^n\Theta(\Delta^N)=\mathbb{H}\). When \(f\) is parabolic with nonzero step, i.e., \(\lim_{n\to\infty}k_{\Delta^N}(f^n(z),f^{n+1}(z))\neq0\) for all \(z\), they show that there exists a holomorphic function \(\Theta:\Delta^N\to\mathbb{H}\) that solves the Abel equation NEWLINE\[NEWLINE\Theta(f(z))=\Theta(z)\pm1NEWLINE\]NEWLINE for all \(z\) and satisfies \(\bigcup_{n\geq0}(\Theta(\Delta^N)\mp n)=\mathbb{H}\).