Spectral Asymptotics for $V$-variable Sierpinski Gaskets

The family of $V$-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals ($V=1$) and random recursive fractals ($V=\infty$). We consider a class of $V$-variable affine nested fractals based on the Sierpinski gasket with a general class of measures. We calculate the spectral exponent for a general measure and find the spectral dimension for these fractals. We show that the spectral properties and on-diagonal heat kernel estimates for $V$-variable fractals are closer to those of scale irregular fractals, in that it is the fluctuations in scale that determine their behaviour but that there are also effects of the spatial variability.