Salem numbers in dynamics of K\"ahler threefolds and complex tori

Abstract

Let $X$ be a compact K\"ahler manifold of dimension $k\leq 4$ 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 $\mbox{dim}(X)=3$ 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.