Generalized $q$-dimensions of measures on nonautonomous fractals

TL;DR

This paper investigates the generalized $q$-dimensions of measures on nonautonomous fractals, providing estimates, formulas, and bounds for various classes of these complex attractors.

Contribution

It introduces new dimension formulas and bounds for measures on nonautonomous fractals, extending classical results to more general and affine nonautonomous sets.

Findings

Derived dimension formulas for measures on nonautonomous attractors.

Established upper bounds for measures on nonautonomous affine sets.

Analyzed variations of affine sets and obtained their $q$-dimension formulas.

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 $.