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

Abstract

We study the spectral dimensions and spectral asymptotics of Krein-Feller operators for arbitrary finite Borel measures on $\left(0,1\right).$ Connections between the spectral dimension, the $L^{q}$-spectrum, the partition entropy and the optimised coarse multifractal dimension are established. In particular, we show that the upper spectral dimension always corresponds to the fixed point of the $L^{q}$-spectrum of the corresponding measure. Natural bounds reveal intrinsic connections to the Minkowski dimension of the support of the associated Borel measure. Further, we give a sufficient condition on the $L^{q}$-spectrum to guarantee the existence of the spectral dimension. As an application, we confirm the existence of the spectral dimension of self-conformal measures with or without overlap as well as of certain measures of pure point type. We construct a simple example for which the spectral dimension does not exist and determine explicitly its upper and lower spectral dimension.