Statistical properties of quadratic polynomials with a neutral fixed point

TL;DR

This paper investigates the statistical behavior of quadratic polynomials with neutral fixed points, demonstrating unique ergodicity and confirming a conjecture about hedgehog dynamics in complex dynamics.

Contribution

It proves the unique ergodicity of quadratic polynomials with neutral fixed points on their measure-theoretic attractors, confirming Perez-Marco's conjecture.

Findings

Maps are uniquely ergodic on their measure-theoretic attractors

The invariant measure is a physical measure for typical orbits

Confirms the conjecture of Perez-Marco on hedgehog dynamics

Abstract

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.