Log-Periodic Asymptotic Expansion of the Spectral Partition Function for Self-Similar Sets

Abstract

Let $${{\fancyscript{Z}}(t)}$$ be the partition function (the trace of the heat semigroup) of the canonical Laplacian on a post-critically finite self-similar set (with uniform resistance scaling factor and good geometric symmetry) or on a generalized Sierpinski carpet. It is proved that $${{\fancyscript{Z}}(t)=\sum_{k=0}^{n}t^{-d_{k}/d_{\mathrm{w}}}G_{k}(-\log t) +O\bigl({\rm exp}(-ct^{-\frac{1}{d_{\mathrm{w}}-1}})\bigr)}$$ as $${t\downarrow 0}$$ for some continuous periodic functions $${G_{k}:{\mathbb{R}}\to{\mathbb{R}}}$$ and $${c\in(0,\infty)}$$ . Here $${d_{\mathrm{w}}\in(1,\infty)}$$ denotes the walk dimension, n = 1 for a post-critically finite self-similar set, n = d for a d-dimensional generalized Sierpinski carpet, $${\{d_{k}\}_{k=0}^{n}\subset[0,\infty)}$$ is strictly decreasing with d n = 0, G 0 is strictly positive and G 1 is either strictly positive or strictly negative depending on the (Neumann or Dirichlet) boundary condition.