Generalized $q$-dimensions of measures on nonautonomous fractals

Abstract: 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 $.