Statistical properties of quadratic polynomials with a neutral fixed point

Abstract

We describe the statistical properties of the dynamics of the quadratic polynomials $P_{\alpha }(z)=e^{2\pi \alpha \mathbf{i}}z+z^2$ on the complex plane, with $\alpha$ of high type. 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 behaviour of typical orbits in the Julia set. This confirms a conjecture of Pérez–Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.