Pre-threshold fractional susceptibility functions at Misiurewicz parameters

TL;DR

This paper proves that certain fractional susceptibility functions for specific unimodal maps at Misiurewicz parameters are holomorphic beyond the unit disk, advancing understanding of their complex analytic properties.

Contribution

It establishes the holomorphic extension of fractional susceptibility functions at Misiurewicz parameters for fractional differentiation indices less than 1/2, addressing a conjecture by Baladi and Smania.

Findings

Fractional susceptibility functions are holomorphic on a disk of radius greater than one.

The result applies to real-analytic unimodal families at Misiurewicz parameters.

It advances the understanding of the complex properties of these functions.

Abstract

We show that the response, frozen and semifreddo fractional susceptibility functions of certain real-analytic unimodal families, at Misiurewicz parameters and for fractional differentiation index $0\le\eta<1/2$, are holomorphic on a disk of radius greater than one. This is a step towards solving a conjecture of Baladi and Smania, in the case of the aforementioned susceptibility functions.