Regularity of the spectrum for expanding maps

TL;DR

This paper extends the differentiability of eigenfunctions associated with the spectrum of weighted transfer operators for expanding maps to the entire discrete spectrum, using resolvent operator regularity.

Contribution

It introduces a method to analyze the regularity of the entire spectrum of transfer operators with respect to parameters, beyond the simple eigenvalue.

Findings

Extended differentiability to the whole discrete spectrum.

Developed a general regularity result for fixed points with loss of regularity.

Applied resolvent operator analysis to spectral regularity.

Abstract

In this short note, we propose to extend differentiability (with respect to a multidimensional parameter) of a normalized eigenfunction associated to the simple, dominating eigenvalue of the weighted transfer operator for a uniformly expanding map, to the whole discrete spectrum. We do so by studying directly the regularity (with respect to the parameter) of the resolvent operator, applying a general regularity result for fixed points of maps having loss of regularity.