On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals

For the canonical heat kernels p _t (x, y) associated with Dirichlet forms on post-critically finite self-similar fractals, e.g. the transition densities (heat kernels) of Brownian motion on affine nested fractals, the non-existence of the limit ${lim_{tdownarrow 0}t^{d_{s}/2}p_{t}(x,x)}$ is established for a “generic” (in particular, almost every) point x, where d _s denotes the spectral dimension. Furthermore the same is proved for any point x in the case of the d-dimensional standard Sierpinski gasket with d ≥ 2 and the N-polygasket with N ≥ 3 odd, e.g. the pentagasket (N = 5) and the heptagasket (N = 7).