Spectral asymptotics of Krein--Feller operators for weak Gibbs measures on self-conformal fractals with overlaps

TL;DR

This paper investigates the spectral properties of Krein--Feller operators on self-conformal fractals with overlaps, establishing the existence of spectral dimensions and asymptotics of eigenvalues for weak Gibbs measures.

Contribution

It extends spectral analysis to fractals with overlaps, linking spectral dimension to the $L^{q}$-spectrum and pressure function, and determines eigenvalue asymptotics under certain regularity conditions.

Findings

Spectral dimension equals the fixed point of the $L^{q}$-spectrum.

Spectral dimension equals the zero of the pressure function under open set condition.

Eigenvalue counting function asymptotics are derived for certain Gibbs measures.

Abstract

We study the spectral dimensions and spectral asymptotics of Krein--Feller operators for weak Gibbs measures on self-conformal fractals with or without overlaps. We show that, restricted to the unit interval, the $L^{q}$-spectrum for every weak Gibbs measure $\rho$ with respect to a $\mathcal{C}^{1}$-IFS exists as a limit. Building on recent results of the authors, we can deduce that the spectral dimension with respect to a weak Gibbs measure exists and equals the fixed point of its $L^{q}$-spectrum. For an IFS satisfying the open set condition, it turns out that the spectral dimension equals the unique zero of the associated pressure function. Moreover, for a Gibbs measure with respect to a $\mathcal{C}^{1+\gamma}$-IFS under the open set condition, we are able to determine the asymptotics of the eigenvalue counting function.