Dimensions of surface repellers and attractors of non-linear planar IFSs
Abstract
We establish that for a Cr-generic surface repeller (1 < r ≤ ∞), its Hausdorff and box dimensions are exactly the unique zero of the sub-additive topological pressure function. As an application of our framework, we show that the attractor of a uniformly non-conformal and weakly irreducible planar non-linear iterated function system (IFS) satisfying the strong separation condition (SSC) attains its expected Hausdorff and box dimensions. Furthermore, as a direct consequence of our generic repeller theorem, we deduce that this dimension formula also holds for a generic Cr planar IFS satisfying the SSC. Using this approach, we also extend the dichotomy result for graphs of Weierstrass-type functions of Ren and Shen (2021) by weakening their real-analytic requirement to arbitrary Cr regularity for r > 1.
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BibTeX
@misc{li2026dimensions,
title = {{Dimensions of surface repellers and attractors of non-linear planar IFSs}},
author = {Jialun Li and Wenyu Pan and Yao Tong and Disheng Xu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30744},
eprint = {2608.30744},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30744},
}
RIS
TY - UNPB
TI - Dimensions of surface repellers and attractors of non-linear planar IFSs
AU - Jialun Li
AU - Wenyu Pan
AU - Yao Tong
AU - Disheng Xu
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30744
UR - https://arxiv.org/abs/2608.30744
AB - We establish that for a C^r-generic surface repeller (1 < r ≤ ∞), its Hausdorff and box dimensions are exactly the unique zero of the sub-additive topological pressure function. As an application of our framework, we show that the attractor of a uniformly non-conformal and weakly irreducible planar non-linear iterated function system (IFS) satisfying the strong separation condition (SSC) attains its expected Hausdorff and box dimensions. Furthermore, as a direct consequence of our generic repeller theorem, we deduce that this dimension formula also holds for a generic C^r planar IFS satisfying the SSC. Using this approach, we also extend the dichotomy result for graphs of Weierstrass-type functions of Ren and Shen (2021) by weakening their real-analytic requirement to arbitrary C^r regularity for r > 1.
ER -