JournalsjemsVol. 20, No. 8pp. 2005–2062

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
Statistical properties of quadratic polynomials with a neutral fixed point cover

A subscription is required to access this article.

Abstract

We describe the statistical properties of the dynamics of the quadratic polynomials Pα(z)=e2παiz+z2P_\alpha(z)=e^{2\pi \alpha i} z+ z^2 on the complex plane, with α\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 P´erez-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