  Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an
algebraically closed field of characteristic zero. We consider the set
$\Sigma_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of
small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in
$\Sigma_{f^{\infty}}$, and the subset $S_f$ of $f$-invariant elements in
$S_{f^{\infty}}$. When $X$ is projective, we prove the finiteness of the set
$P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an
optimal upper bound (of cardinality) $$\sharp P_{f^n}\le d_1(f)^n(1+o(1))$$ as
$n\to \infty$, where $d_1(f)$ is the first dynamic degree of $f$. When $X$ is
an algebraic group (with $f$ being a translation of an isogeny), or a (not
necessarily complete) toric variety (with $f$ stabilizing the big torus), we
give an optimal upper bound $$\sharp S_{f^n}\le
d_1(f)^{n\cdot\dim(X)}(1+o(1))$$ as $n \to \infty$, which slightly generalizes
a conjecture of S.-W. Zhang for polarized $f$.
