  We describe the statistical properties of the dynamics of the quadratic
polynomials P_a(z):=e^{2\pi a i} z+z^2 on the complex plane, with a of high
return times. In particular, we show that these maps are uniquely ergodic on
their measure theoretic attractors, and the unique invariant probability is a
physical measure describing the statistical behavior of typical orbits in the
Julia set. This confirms a conjecture of Perez-Marco on the unique ergodicity
of hedgehog dynamics, in this class of maps.
