Dynamics of rational surface automorphisms: rotation domains (Q2881358)

Given \(f\) a biholomorphic automorphism of a compact complex surface \(X\), the regular part of its dynamics occurs on the Fatou set of \(f\), where the forward iterates are equicontinuous. A Fatou component, that is a connected component \(U\) of the Fatou set, is called a ``rotation domain of rank \(d\)'' if \(f^p(U)\subset U\) for some \(p\geq 1\) and \(f^p|_U\) generates a (real torus) \(\mathbb{T}^d\)-action on \(U\). In dimension \(1\), rotation domains correspond to Siegel discs or Herman rings.NEWLINENEWLINEIn the paper under review, the authors consider surface automorphisms such that the induced map \(f^*\) on \(H^2(X, \mathbb{R})\) has an eigenvalue greater than one, which is equivalent to the condition of positive entropy for \(f\), see [\textit{S. Cantat}, Acta Math. 187, No. 1, 1--57 (2001; Zbl 1045.37007)]. The authors show that a positive-entropy rational surface automorphism can have a ``large'' rotation domain, i.e., a rotation domain containing both a curve of fixed points and isolated fixed points.NEWLINENEWLINEMore precisely, the authors use birational maps of the plane which are defined, in affine coordinates \((x,y)\), by NEWLINE\[NEWLINE f(x,y) = (y, -\delta x + cy + y^{-1}), NEWLINE\]NEWLINE where \(\delta\) is a root of modulus 1 of the polynomial NEWLINE\[NEWLINE \chi_{n,m}(t) = \frac{t (t^{nm} -1) (t^n -2t^{n-1} + 1)}{(t^n - 1)(t-1)} + 1, NEWLINE\]NEWLINE and \(c= 2\sqrt{\delta}\cos(j\pi/n)\), for \(1\leq j\leq n-1\) and \((j,n)=1\). They construct a complex algebraic surface \(\pi: X\to \mathbb{P}^2\) by performing iterated blowups to level 3. They show that with their choice for \(c\) and \(\delta\), the induced map \(f_X=\pi^{-1}\circ f\circ \pi\) is an automorphism of \(X\), of positive entropy. Moreover the iterates of \(f\) have rotation domains containing a curve of fixed points, as well as invariant curves, and they are union of invariant (Siegel) discs on which the iterates act as irrational rotations. The existence of rotation domains can be given by linearization. However, since linearization is a local technique, to understand the global nature of the Fatou component, the authors introduce a global model, where the rotation domain can be globally linearized.