Abstract. On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy E ϕ and conformal Laplacian ∆ ϕ for a given conformal factor ϕ, based on the corresponding notions in Riemannian geometry in dimension n = 2. We derive a differential equation that describes the dependence of the effective resistances of E ϕ on ϕ. We show that the spectrum of ∆ ϕ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function ϕ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the L p dimensions of these measures for integer values of p ≥ 2. We derive analogous linear extension algorithms for energy measures on related fractals.

We study the wave equation for a stationary Lorentzian metric in the case of two space dimensions. Assuming that the metric has a singularity of the appropriate form, surrounded by an ergosphere which is a smooth Jordan curve, we prove the existence of a black hole with the boundary (called the event horizon) that is piece-wise smooth, generally having corners. We consider a physical model of acoustic black hole whose event horizon has corners. In the end of the paper we consider the determination of a black hole by the boundary measurements.

Abstract. We study a semiclassical inverse spectral problem based on a spectral asymptotics result of [13], which applies to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. The eigenvalues in a suitable complex window have an expansion in terms of a quantum Birkhoff normal form (QBNF) for the operator near several Lagrangian tori that are invariant under the classical dynamics and satisfy a Diophantine condition. In this work, we prove that the normal form near a single Diophantine torus is uniquely determined by the associated eigenvalues. We also discuss the normalization procedure and symmetries of the QBNF near a Diophantine torus.

We study the distribution of eigenvalues for selfadjoint h-pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength ε of the perturbation is ≪ h, the spectrum displays a cluster structure, and assuming that ε ≫ h 2 (or sometimes ≫ h N 0 , for N 0 > 1 large), we obtain a complete asymptotic description of the individual eigenvalues inside subclusters, corresponding to the regular values of the leading symbol of the perturbation, averaged along the flow.