Pre-threshold fractional susceptibility function: holomorphy and response formula

TL;DR

This paper proves the holomorphy of the fractional susceptibility function for certain dynamical systems and establishes a fractional response formula, advancing understanding of statistical properties in complex systems.

Contribution

It constructs parameter sets with exponential mixing and proves holomorphy of the fractional susceptibility function for the logistic family, confirming a recent conjecture.

Findings

Holomorphy of the fractional susceptibility function in a disk larger than one.

Construction of parameter sets with exponential mixing.

Validation of the fractional response formula.

Abstract

For certain smooth unimodal families with negative Schwarzian derivative, we construct a set of Collet-Eckmann and subexponentially recurrent parameters $\Omega$, whose complement set has sufficiently fast decaying density, on which exponential mixing with uniform rates occurs. We use this construction to establish holomorphy of the true fractional susceptibility function of the logistic family, in a disk of radius larger than one, for differentiation index $0\le\eta<1/2$, as recently conjectured by Baladi and Smania. We also obtain a fractional response formula.