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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 \({\lim_{t\downarrow 0}t^{d_{s}/2}p_{t}(x,x)}\) 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).

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