Statistical properties of quadratic polynomials with a neutral fixed point

  • Artur Avila

    Institut de Mathématiques de Jussieu, Paris Rive Gauche, France, and IMPA, Rio de Janeiro, Brazil
  • Davoud Cheraghi

    Imperial College London, UK

Abstract

We describe the statistical properties of the dynamics of the quadratic polynomials on the complex plane, with 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.

Cite this article

Artur Avila, Davoud Cheraghi, Statistical properties of quadratic polynomials with a neutral fixed point. J. Eur. Math. Soc. 20 (2018), no. 8, pp. 2005–2062

DOI 10.4171/JEMS/805