Int-amplified endomorphisms of compact K\"ahler spaces

Let $X$ be a normal compact K"ahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^{\ast }\xi -\xi =\eta$ for some K"ahler classes $\xi$ and $\eta$. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact K"ahler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a positive power, we can run the minimal model program (MMP) $f$-equivariantly for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of Picard number one.