  We consider dynamical systems given by interval maps with a finite number of
turning points (including critical points, discontinuities) possibly of
different critical orders from two sides. If such a map $f$ is continuous and
piecewise $C^2$, satisfying negative Schwarzian derivative and some summability
conditions on the growth of derivatives and recurrence along the turning
orbits, then $f$ has finitely many attractors whose union of basins of
attraction has total probability, and each attractor supports an absolutely
continuous invariant probability measure $\mu$. Over each attractor there
exists a renormalization $(f^m,\mu)$ that is exact, and the rates of mixing
(decay of correlations) are strongly related to the rates of growth of the
derivatives and recurrence along the turning orbits in the attractors. We also
give a sufficient condition for $(f^m,\mu)$ to satisfy the Central Limit
Theorem. In some sense, we give a fairly complete global picture of the
dynamics of such maps. Similarly, we can get similar statistical properties for
interval maps with critical points and discontinuities under some more
assumptions.
