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In the framework of the Laplacian transport, described by a Robin boundaryvalue problem in an exterior domain in $\mathbb{R}^n$, we generalize thedefinition of the Poincar\'e-Steklov operator to $d$-set boundaries, $n-2<d<n$, and give its spectral properties to compare to the spectra of theinterior domain and also of a truncated domain, considered as an approximationof the exterior case. The well-posedness of the Robin boundary value problemsfor the truncated and exterior domains is given in the general framework of$n$-sets. The results are obtained thanks to a generalization of the continuityand compactness properties of the trace and extension operators in Sobolev,Lebesgue and Besov spaces, in particular, by a generalization of the classicalRellich-Kondrachov Theorem of compact embeddings for $n$ and $d$-sets.

Dirichlet-to-Neumann or Poincar\'e-Steklov operator on fractals described by d -sets