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Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media

by H. Ammari and Y.T. Chow and F. Han

(Report number 2026-30)

Abstract
We develop the first rigorous topological framework for an auxiliary boundary-integral model motivated by physical scaling identities for wave scattering from self-similar or fractal media. The model is periodic in logarithmic scale, and a multiplicative Bloch-Floquet-Zak transform fiberizes its interscale coupling. For sufficiently small, well-separated components, equilibrium densities define a computable finite-dimensional projected matrix, while Riesz projections select the corresponding invariant spectral subspaces of the full boundary symbol. Under suitable spectral-isolation and point-gap conditions, and after recentering the two families at their respective same-scale reference values, we prove that the determinant winding of the exact Riesz-reduced family agrees with that of the projected matrix family. For regular-simplex configurations, we derive explicit nonzero winding formulas and track the resulting local winding data across finite prefractal levels and along a geometric sequence of wavenumbers. We also formulate conditional winding data for multiple dilation centers and show that the Zak phase of the chiral Hermitianization equals \(\pi\) times the point-gap winding modulo \(2\pi\). Thus, the scale-periodic boundary model admits a rigorous topological reduction.

Keywords: Self-similar medium, wave scattering, Bloch-Floquet theory, topological invariant.

@Techreport{ACH26_1185,
  author = {H. Ammari and Y.T. Chow and F. Han},
  title = {Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-30.pdf },
  year = {2026}
}

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