Rigidity of rationally connected smooth projective varieties from dynamical viewpoints

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 ^∼= (P¹)^×n if and only if X admits a surjective endomorphism f such that the eigenvalues of f^∗|_N1(X) (without counting multiplicities) are n distinct real numbers greater than 1.