Salem numbers in dynamics on Kähler threefolds and complex tori

Abstract

Let $X$ be a compact Kähler 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.