  Let $X$ be a rationally connected smooth projective variety of dimension $n$.
We show that $X$ is a toric variety if and only if $X$ admits an int-amplified
endomorphism with totally invariant ramification divisor. We also show that
$X\cong (\mathbb{P}^1)^{\times n}$ if and only if $X$ admits a surjective
endomorphism $f$ such that the eigenvalues of $f^*|_{\text{N}^1(X)}$ (without
counting multiplicities) are $n$ distinct real numbers greater than $1$.
