  Let $\mathbb{K}$ be an algebraically closed field, $X$ a smooth projective
variety over $\mathbb{K}$ and $f:X\rightarrow X$ a dominant regular morphism.
Let $N^i(X)$ be the group of algebraic cycles modulo numerical equivalence. Let
$\chi (f)$ be the spectral radius of the pullback
$f^*:H^*(X,\mathbb{Q}_l)\rightarrow H^*(X,\mathbb{Q}_l)$ on $l$-adic cohomology
groups, and $\lambda (f)$ the spectral radius of the pullback
$f^*:N^*(X)\rightarrow N^*(X)$. We prove in this paper, by using consequences
of Deligne's proof of Weil's Riemann hypothesis, that $\chi (f)=\lambda (f)$.
This answers affirmatively a question posed by Esnault and Srinivas.
Consequently, the algebraic entropy $\log \chi (f)$ of an endomorphism is both
a birational invariant and \'etale invariant. More general results are proven
if either $\mathbb{K}=\overline{\mathbb{F}_p}$ or the Fundamental Conjecture D
(numerical equivalence vs homological equivalence) holds. Among other results
in the paper, we show that if some properties of dynamical degrees, known in
the case $\mathbb{K}=\mathbb{C}$, hold in positive characteristics, then simple
proofs of Weil's Riemann hypothesis follow.
