Generalized q-dimensions of measures on Non-autonomous conformal sets

TL;DR

This paper investigates the generalized q-dimensions of measures on non-autonomous conformal attractors, establishing bounds, formulas under separation conditions, and extensions to autonomous sets, advancing fractal dimension theory.

Contribution

It provides new bounds and explicit formulas for generalized q-dimensions of measures on non-autonomous conformal sets, including special cases like Bernoulli measures and autonomous sets.

Findings

Critical values of pressure functions bound the q-dimensions.

Dimension formulas are derived under separation conditions.

Simplified formulas for Bernoulli measures and autonomous sets.

Abstract

We study the generalized q-dimensions of measures supported on non-autonomous conformal attractors, which are the generalizations of Moran sets and the attractors of iterated function systems. We first prove that the critical values of generalized upper and lower pressure functions are always the upper bounds for the upper and lower generalized q-dimensions of measures supported on non-autonomous conformal sets. Then we obtain dimension formulas for generalized q-dimensions if non-autonomous conformal attractors satisfy certain separation conditions, and moreover, the generalized q-dimension formulae may be simplified for the Bernoulli measures. Finally, we provide the generalized q-dimension formulae for measures supported on autonomous conformal sets.