  Let $X$ be a smooth projective variety over an algebraically closed field of
arbitrary characteristic, and $f$ a dynamical correspondence of $X$. In 2016,
the second author conjectured that the dynamical degrees of $f$ defined by the
pullback actions on \'etale cohomology groups and on numerical cycle class
groups are equivalent, which we call the dynamical degree comparison (DDC)
conjecture. It contains the generalized Weil's Riemann hypothesis (for
polarized endomorphisms) as a special case.
  To proceed, we introduce the so-called Conjecture $G_r$, which is a
quantitative strengthening of the standard conjecture $C$ and holds on abelian
varieties and Kummer surfaces. We prove that for arbitrary varieties,
Conjecture $G_r$ yields the generalized Weil's Riemann hypothesis. Moreover,
Conjecture $G_r$ plus the standard conjecture $D$ imply the so-called norm
comparison (NC) conjecture, whose consequences include the DDC conjecture and
the generalized semisimplicity conjecture (for polarized endomorphisms). As an
application, we obtain new results on the DDC conjecture for abelian varieties
and Kummer surfaces, and the generalized semisimplicity conjecture for Kummer
surfaces. Finally, we also obtain a similar comparison result for effective
finite correspondences of abelian varieties.
