  We study rational functions satisfying summability conditions - a family of
weak conditions on the expansion along the critical orbits. Assuming their
appropriate versions, we derive many nice properties:
  There exists a unique, ergodic, and non-atomic conformal measure with
exponent equal to the Hausdorff dimension of the Julia set.
  There is an absolutely continuous invariant measure with respect to this
conformal measure.
  The Minkowski dimension of the Julia set is strictly less than 2.
  Either the Julia set is the whole sphere, or the dynamics is unstable.
  For such polynomials and Blaschke products the Julia set is conformally
removable.
  The connected components of the boundary of invariant Fatou components are
locally connected.
  Finally, we derive a conformal analogue of Jakobson-Benedicks-Carleson
theorem and prove the external continuity of the Hausdorff dimension of Julia
sets for almost all points in the Mandelbrot set with respect to the harmonic
measure.
  Some of the results extend to the multimodal maps of an interval.
