Iterated relation systems on Riemannian manifolds

TL;DR

This paper introduces iterated relation systems on Riemannian manifolds to analyze fractals that are not well-handled by traditional iterated function systems, providing new dimension formulas under certain conditions.

Contribution

It develops the concept of iterated relation systems and links their attractors to graph-directed systems, improving previous methods for fractal dimension calculation.

Findings

Established conditions for attractor identification with graph-directed systems

Derived dimension formulas under graph open set and finite type conditions

Enhanced analysis of fractals on Riemannian manifolds

Abstract

For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we introduce the notion of iterated relation systems. We study the attractor of an iterated relation system and formulate a condition under which such an attractor can be identified with that of an associated graph-directed iterated function system. Using this method, we obtain dimension formulas for the attractor of an iterated relation system under the graph open set condition or the graph finite type condition. This method improves the one in [Ngai-Xu, J. Geom. Anal. {\bf 33} (2023), 262], which relies on knowing the specific structure of the attractor.