  We first characterize those composition operators that are essentially normal
on the weighted Bergman space $A^2_s(D)$ for any real $s>-1$, where induced
symbols are automorphisms of the unit disk $D$. Using the same technique, we
investigate the automorphic composition operators on the Hardy space $H^2(B_N)$
and the weighted Bergman spaces $A^2_s(B_N)$ ($s>-1$). Furthermore, we give
some composition operators induced by linear fractional self-maps of the unit
ball $B_N$ that are not essentially normal.
