# The qth packing moments and the packing Lq-spectra of directed graph self-similar measures

## TL;DR

This paper analyzes the packing moments and Lq-spectra of self-similar measures generated by directed graph iterated function systems, providing explicit formulas for their behavior at small scales using renewal theory.

## Contribution

It introduces explicit calculations for the qth packing moments and spectra of self-similar measures under various separation conditions, extending previous results to more general settings.

## Key findings

- Explicit power law behavior of packing moments at small scales
- Determination of packing Lq-spectra and convergence rates
- Applicability to systems satisfying SSC and OSC conditions

## Abstract

Any self-similar directed graph iterated function system with probabilities, defined on m-dimensional Euclidean space, determines a unique list of self-similar Borel probability measures whose supports are the components of the attractor. Using an application of the Renewal Theorem we obtain an explicit calculable value for the power law behaviour of the qth packing moments of the self-similar measures at scale r as r tends to 0 in the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any real q if the strong separation condition (SSC) holds, (ii) for non-negative q if the weaker open set condition (OSC) holds, where we also assume that a non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the packing Lq-spectra and their exact rate of convergence to be determined.