Dynamics on \(K3\) surfaces: Salem numbers and Siegel disks (Q2782023)

The main result of the paper states that there exist holomorphic automorphisms of \(K3\) surfaces with Siegel disks (a Siegel disk of an automorphism \(f\) of \(X\) is a domain \(V\subset X\) such that \(f(V)=V\) and \(f| V\) is analytically conjugate to an irrational rotation of the bidisk). There are at most countably many such submorphisms, up to isomorphism, and there are no Siegel disks on projective \(K3\) surfaces. To construct a Siegel disk, the author takes a linear transformation \(F\) of \(\mathbb Z[y]/S(y)\) with a certain lattice structure, where \(S(y)\) is the minimal polynomial for a Salem number \(\lambda>1\) of degree 22 and trace \(-1\). The lattice is isomorphic to \(H^2(X,\mathbb{Z})\) for a \(K3\) surface \(X\), and there exists \(f\in\Aut(X)\) such that \(f^*:H^2 (X,\mathbb Z)\to H^2 (X,\mathbb Z)\) is conjugate to \(F\). Using the Atiyah-Bott fixed point theorem and results from transcendence theory, it is then shown that \(f\) has a Siegel disk. Explicit examples of such automorphisms are given.