Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps

TL;DR

This paper proves that for a holomorphic family of quadratic-like Collet-Eckmann maps, the integral of any real analytic function against the SRB measure varies real analytically with the parameter.

Contribution

It establishes the real analyticity of the SRB measure's integral with respect to parameters in a family of quadratic-like maps under Collet-Eckmann conditions.

Findings

The map t -> ∫ g dm_t is real analytic for real analytic g.

The result applies to families with all periodic orbits repelling.

It extends understanding of statistical stability in complex dynamical systems.

Abstract

We show that if f_t is a holomorphic family of quadratic-like maps with all periodic orbits repelling so that for each real t the map f_t is a real Collet-Eckmann S-unimodal map then, writing m_t for the unique absolutely continuous invariant probability measure of f_t, the map t -> \int g dm_t is real analytic for any real analytic function g.