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The main object of this work is the top-dimensional Laplacian operator of asimplicial complex $K$. We study its spectral limiting behavior under a givennon-trivial subdivision procedure $\text{div}$. It will be shown that in case$\text{div}$ satisfies a property we call inclusion-uniformity its spectrumconverges to a universal limiting distribution only depending on the dimensionof $K$. This class of subdivisions contains important special cases such as theedgewise subdivision $\text{esd}_r$ for $r\geq 2$ and dimension $d=2$ or thebarycentric subdivision $\text{sd}$. This parallels a result of Brenti andWelker showing that the roots of $f$-polynomials of iterated barycentricsubdivisions converge to a universal set of roots only depending on thedimension of $K$.  Furthermore we determine the family of universal limiting functions for theparticular subdivision where the top dimensional faces are replaced by a coneover their boundary. We will show that this choice of $\text{div}$ is thenatural generalization of graph subdivision in the spectral sense. These limitsare obtained by explicit spectral decimation of the sequence of its dual graphswhich is represented as a sequence of Schreier graphs on a rooted regular tree.  Finally we will point out that a generic sequence of iterated subdivisionscan be realized by a sequence of graphs as in spectral analysis on fractals. Wewill give a construction of a self-similar sequence of graphs which dualizesthe iterated application of subdivision.

Universal Limit Theorem for Spectra of iterated inclusion-uniform Subdivisions