Iterated Relation Systems on Riemannian Manifolds
Abstract
1. Introduction
2. Statement of Main Results
- (a)
- There exists an integer such that for any and any ,
- (b)
- For any , if , we require that the following conditions are satisfied:
- (i)
- can be decomposed as a finite family of contractions , where , and can be decomposed as a finite family of contractions , where .
- (ii)
- There exists a good partition on with respect to some finite family of contractions decomposed from (not necessarily the one in ).
- (a)
- , where α is the unique real number such that .
- (b)
- In particular, let be a minimal simplified GIFS associated with G and assume that consists of contractive similitudes . Suppose that . If satisfies (OSC), then is the unique number α satisfyingwhere is the contraction ratio of .
3. Iterated Relation Systems
- (a)
- for any nonempty compact set and , is a nonempty compact set;
- (b)
- for any , .
- (C)
- For any , if there exist some , a subsequence of , and satisfying for all , then there exists a subsequence of converging to some satisfying .
- (a)
- .
- (b)
- If Condition (C) holds, then , and thus (1) holds.
4. Associated Graph-Directed Iterated Function Systems
- (a)
- Assume that conditions (i) and (ii) below hold.
- (i)
- For any and ,
- (ii)
- For any , there exists such that for each and , there exists a contraction , and the following holds
Then we say that is a finite family of contractions decomposed from and that each is a branch of . - (b)
- Let be a collection of compact subsets of satisfyingLet be a finite family of contractions decomposed from and composed of all the branches of . Fix . Assume that for any and , is invariant under for all . Then we call a good partition of with respect to .
- (1)
- , where and .
- (2)
- , where and .
- (3)
- is invariant under , and is invariant under , where .
- (a)
- There exists an integer such that for any and any ,
- (b)
- For any , if , we require that the following conditions are satisfied.
- (i)
- can be decomposed as a finite family of contractions , where , and can be decomposed as a finite family of contractions , where .
- (ii)
- There exist or such that for any and any ,and for any ,
- (a)
- , where and ;
- (b)
- , where and .
- (a)
- and , where .
- (b)
- Let be contractions associated with , and let be an invariant family under . Then for any ,
5. Hausdorff Dimension of Graph Self-Similar Sets Without Overlaps
- (a)
- ;
- (b)
- , for all distinct and .
6. Hausdorff Dimension of Graph Self-Similar Sets with Overlaps
6.1. Graph Finite Type Condition
- (a)
- In (29), .
- (b)
- For and satisfying , and for any integer , a directed path satisfies if and only if it satisfies .
6.2. Examples
- (1)
- There does not exist a global bi-Lipschitz map such that .
- (2)
- Let . One may try to construct a bi-Lipschitz map , where . As might not be a self-similar set, it is not clear how to compute the Hausdorff dimension of .
- (3)
- If K is has a neighborhood on which there exists a local isometry mapping U to , then the problem is much easier; however, this is not the case for Example 2. In fact, although K can be covered by finitely or countably many coordinate charts and then pulled into the Euclidean space by isometries, the image of K in lacks a well-defined structure, and it is not clear how to find an IFS that generates it.
7. IRSs on Riemannian Manifolds That Are Not Locally Euclidean
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Some Examples
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Liu, J.; Ngai, S.-M.; Ouyang, L. Iterated Relation Systems on Riemannian Manifolds. Fractal Fract. 2025, 9, 637. https://doi.org/10.3390/fractalfract9100637
Liu J, Ngai S-M, Ouyang L. Iterated Relation Systems on Riemannian Manifolds. Fractal and Fractional. 2025; 9(10):637. https://doi.org/10.3390/fractalfract9100637
Chicago/Turabian StyleLiu, Jie, Sze-Man Ngai, and Lei Ouyang. 2025. "Iterated Relation Systems on Riemannian Manifolds" Fractal and Fractional 9, no. 10: 637. https://doi.org/10.3390/fractalfract9100637
APA StyleLiu, J., Ngai, S.-M., & Ouyang, L. (2025). Iterated Relation Systems on Riemannian Manifolds. Fractal and Fractional, 9(10), 637. https://doi.org/10.3390/fractalfract9100637