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

Marc Kesseböhmer, Aljoscha Niemann

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.

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 Connections between the spectral dimension, the -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 -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 -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.

Paper Structure

This paper contains 17 sections, 44 theorems, 239 equations, 2 figures.

Table of Contents

  1. Introduction and statement of main results
  2. Spectral theory for Kreı̆n-Feller operators
  3. Form approach
  4. Spectral problem, min-max principle, and sub- and superadditivity
  5. Neumann boundary conditions
  6. Optimal partitions and partition entropy
  7. Basic observations
  8. Some global bounds
  9. $L^{q}$-spectrum and coarse multifractal formalism
  10. The $L^{q}$-spectrum
  11. Upper bounds in terms of the $L^{q}$-spectrum
  12. Lower bounds in terms of the optimize coarse multifractal dimension
  13. Proof of main results
  14. Examples
  15. $C^{1}$-self-conformal measures
  16. ...and 2 more sections

Key Result

Theorem 1.1

For all $1<m\leq3$, we have In particular, $\overline{q}_{\nu}\leq1/2$ and the following necessary and sufficient conditions hold for the spectral dimension to exist: We would like to emphasize that if the spectral dimension exists, then it is given by purely measure-geometric data, which is encoded in the $\nu$-partition entropy $\overline{h}_{\nu}=\underline{h}_{\nu}$. We call $\nu$regular, if

Figures (2)

  • Figure 4.1: Moment generating function $\beta_{\nu}$ for $\nu$ with (dashed) tangent to $\beta_{\nu}$ of slope $-\alpha$ in $\left(q,\beta_{\nu}\left(q\right)\right)$ which intersects the $y$-axis in its Legendre transform $\widehat{\beta}_{\nu}\left(\alpha\right)$ and the bisector of the first quadrant in $\widehat{\beta}_{\nu}\left(\alpha\right)/\left(1+\alpha\right)$. Here $\nu$ is chosen to be the $\left(0.05,0.95\right)$-Salem measure with full support $\mathop{\mathrm{supp}}\nolimits\nu=[0,1]$. The intersection of $\beta_{\nu}$ with the $y$-axis gives the Minkowski dimension of $\mathop{\mathrm{supp}}\nolimits\nu$, namely $1$, and the intersection with the (dotted) tangent to $\beta_{\nu}$ in $\left(0,1\right)$ gives the Hausdorff dimension $\dim_{H}\left(\nu\right)$ of the measure which equals $\left(0.05\log\left(0.05\right)+0.95\log\left(0.95\right)\right)/\log\left(2\right)$. The spectral dimension $s_{\nu}$ is equal to the maximum over all $\widehat{\beta}_{\nu}\left(\alpha\right)/\left(1+\alpha\right)$ given by $\overline{q}_{\nu}$.
  • Figure 5.1: $\beta_{\nu^{(\xi)}}$ and $\liminf\beta_{n}^{\nu^{(\xi)}}$with $p_{1}=0.25$.

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Corollary 1.9
  • Proposition 1.10
  • ...and 97 more