On surjective $\alpha$-amplified endomorphisms of projective varieties
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Abstract
Let $X$ be a projective variety of dimension $d$ over a field ${\bf k}$, and let $f:X\rightarrow X$ be a surjective endomorphism of $X$. In this paper, we prove that for any positive real number $\alpha$, any surjective, but non-isomorphic, $\alpha$-amplified endomorphism $f$ of a projective variety $X$ is of positive entropy and its first dynamical degree $\lambda_1(f)$ is not equal to $\alpha$. We also prove that for any real number $\alpha$ greater than or equal to $1$ any surjective $\alpha$-amplified endomorphism $f$ of a projective variety $X$ with positive entropy such that the set of periodic points of $f$ is Zariski dense in $X$ should be always PCD. To be more precise, we show that, when the set of all periodic points of $f_K$ is Zariski dense in $X_K$ for some uncountable algebraically closed field extension $K$ of ${\bf k}$, the set is actually countable.
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