# Dimensions and dimension spectra of Non-autonomous iterated function systems

## TL;DR

This paper investigates the intermediate dimension spectra of non-autonomous conformal sets, unifying various fractal dimensions, and provides formulas for their Hausdorff, packing, and box dimensions, including for systems with countably many mappings.

## Contribution

It introduces a formula for the intermediate dimension spectra of non-autonomous conformal sets using topological pressures, extending understanding of their fractal dimensions.

## Key findings

- Derived the intermediate dimension spectra formula using topological pressures.
- Simplified formulas for Hausdorff, packing, and box dimensions of these sets.
- Established Hausdorff dimension formulas for systems with countably many conformal mappings.

## Abstract

Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is called a non-autonomous conformal set. In this paper, we study intermediate dimension spectra of non-autonomous conformal sets which provide a unifying framework for Hausdorff and box-counting dimensions. First, we obtain the intermediate dimension spectra formula of non-autonomous conformal sets by using upper and lower topological pressures. As a consequence, we obtain simplified forms of their Hausdorff, packing and box dimensions. Finally, we explore the Hausdorff dimensions of the non-autonomous infinite conformal iterated function systems which consists of countably many conformal mappings at each level, and we provide the Hausdorff dimension formula under certain conditions.