If the symbolic local dimension is >1 and the perturbation has Fourier decay, the random self-similar measure has an almost-surely Hölder continuous density, implying the random fractal contains an interior point almost surely.
Generalized $q$-dimensions of measures on nonautonomous fractals
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In the paper, we study the generalized $q$-dimensions of measures supported by nonautonomous attractors, which are the generalization of classic Moran sets and attractors of iterated function systems. First, we estimate the generalized $q$-dimensions of measures supported on nonautonomous attractors, and we provide dimension formulas for generalized $q$-dimensions of measures supported on nonautonomous similar attractor under certain separation conditions. Next, we investigate the generalized $q$-dimensions of measures supported on nonautonomous affine sets and obtain the upper bounds. Finally, we study two variations of nonautonomous affine sets and obtain their dimension formulas for $q\geq 1 $.
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Smoothness of random self-similar measures on the line and the existence of interior points
If the symbolic local dimension is >1 and the perturbation has Fourier decay, the random self-similar measure has an almost-surely Hölder continuous density, implying the random fractal contains an interior point almost surely.