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

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.

31 pages, 1 figure

References in corpus (1)

• Spectral dimensions of Krein--Feller operators and $L^q$-spectra

Cited by in corpus (2)

• Quantization dimensions of compactly supported probability measures via Rényi dimensions
• Approximation order of Kolmogorov diameters via $L^q$-spectra and applications to polyharmonic operators