Salem numbers in dynamics on Kähler threefolds and complex tori

Abstract

Let \(X\) be a compact Kahler manifold of dimension \(k\!\le \! 4\) and \(f{:}X\!\rightarrow \! X\) a pseudo-automorphism. If the first dynamical degree \(\lambda _1(f)\) is a Salem number, we show that either \(\lambda _1(f)=\lambda _{k-1}(f)\) or \(\lambda _1(f)^2=\lambda _{k-2}(f)\). In particular, if \({\dim }(X)=3\) then \(\lambda _1(f)=\lambda _2(f)\). We use this to show that if \(X\) is a complex 3-torus and \(f\) is an automorphism of \(X\) with \(\lambda _1(f)>1\), then \(f\) has a non-trivial equivariant holomorphic fibration if and only if \(\lambda _1(f)\) is a Salem number. If \(X\) is a complex 3-torus having an automorphism \(f\) with \(\lambda _1(f)=\lambda _2(f)>1\) but is not a Salem number, then the Picard number of \(X\) must be 0, 3 or 9, and all these cases can be realized.