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

TL;DR

This work analyzes spectral properties of Krein–Feller operators $\Delta_{\rho}$ for weak Gibbs measures $\rho$ on self-conformal fractals, allowing overlaps. It develops a Dirichlet-form framework for generalized Krein–Feller operators and links spectral data to thermodynamic quantities via the $L^{q}$-spectrum $\beta_{\rho}$ and the pressure function $P$, establishing when the spectral dimension exists and how it is computed. The main contributions show that the spectral dimension $s_{\rho|_{(0,1)}}$ equals the fixed point $q_{\rho}$ of $\beta_{\rho}$, and under OSC this reduces to the zero $z_{\rho}$ of $P$, with sharp power-law asymptotics $N_{\rho}(x)\asymp x^{z_{\rho}}$ for Hölder potentials on $C^{1+\gamma}$-IFS. The results extend to overlapped and nonlinear fractals by combining Barral–Feng/Barral methods with recent nonlinear-thermodynamic insights, linking spectral geometry to thermodynamic data on self-conformal sets.

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 $ρ$ 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 $\mathcal{C}^{1+γ}$-IFS under the open set condition, we are able to determine the asymptotics of the eigenvalue counting function.

Figures (1)

• Figure 2.1: The graph of $\beta _{\varrho }$ (solid line) for a dimensionally regular IFS with four contraction ratios equal to $1/2$ and an associated self-similar measure $\varrho$ with probability vector $\left (0.001,0.001,0.05,0.948\right )$. The graph of $\beta _{\varrho }$ coincides on $\left [\widetilde{q},1\right ]$ with $\tau$ (dotted line) as defined in (\ref{['eq:Def_tau']}) and we have $\tau \left (0\right )=2$. The linear part of $\beta _{\varrho }$ is determined by the tangent to the graph of $\tau$ over the positive $x$-axis through the point $\left (0,1\right )$. The intersection with the dashed line with slope $1$ gives the value for the spectral dimension $s_{\varrho }$.

Theorems & Definitions (58)

• Theorem 1.1
• Corollary 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7: Spectral asymptotics
• Lemma 2.1
• proof
• Proposition 2.2