Work

Iterated Relation Systems on Riemannian Manifolds

Year: 2025

Type: article

Abstract: For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as these fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we introduce the novel 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 on locally Euclidean Riemannian manifolds, under the graph open set condition or the graph finite type condition. This method improves the one by Ngai and Xu, which relies on knowing the specific structure of the attractor. We also study fractals generated by iterated relation systems on Riemannian manifolds that are not locally Euclidean.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract9100637

Authors: Jie Liu, Sze-Man Ngai, Lei Ouyang

Institutions: Hunan Normal University, Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), Beijing Institute of Mathematical Sciences and Applications

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funders: National Natural Science Foundation of China, Construct Program of the Key Discipline in Hunan Province, Division of Mathematical Sciences