Dimension results for inhomogeneous Moran set constructions

TL;DR

This paper calculates various fractal dimensions for complex inhomogeneous Moran set constructions, extending classical theories by allowing nonlinear, weakly conformal, and topologically intricate basic sets, with applications to stochastic systems.

Contribution

It introduces a method to compute fractal dimensions for inhomogeneous Moran sets with nonlinear, weakly conformal contractions and complex topologies, beyond classical iterated function systems.

Findings

Computed Hausdorff, box, and packing dimensions for these sets.

Extended dimension theory to include stochastic and chaotic control of contractions.

Applied thermodynamic formalism to derive fractal dimensions.

Abstract

We compute the Hausdorff, upper box and packing dimensions for certain inhomogeneous Moran set constructions. These constructions are beyond the classical theory of iterated function systems, as different nonlinear contraction transformations are applied at each step. Moreover, we also allow the contractions to be weakly conformal and consider situations where the contraction rates have an infimum of zero. In addition, the basic sets of the construction are allowed to have a complicated topology such as having fractal boundaries. Using techniques from thermodynamic formalism we calculate the fractal dimension of the limit set of the construction. As a main application we consider dimension results for stochastic inhomogeneous Moran set constructions, where chaotic dynamical systems are used to control the contraction factors at each step of the construction.