Asymptotic separation of periodic orbits and fractal dimension

Abstract

For a totally bounded metric dynamical system $(X, d, T)$ we introduce a new critical value s(X, d, T) which quantifies the asymptotic separation of periodic orbits. More precisely, s(X,d,T) is defined to be the supremum of all s≥ 0 for which there exists a sequence of periodic orbits {𝒪_k}_k=1^∞ in (X,d,T) such that lim_{k→ ∞} #𝒪_k = +∞ and liminf_k→∞#𝒪_k·η(𝒪_k)^s >0, where η(𝒪_k) denotes the smallest distance between distinct points in 𝒪_k. When the dynamical system is induced by a self-similar iterated function system ℱ satisfying the strong separation condition, we prove that this critical value is equal to the Hausdorff dimension s_ℱ of its self-similar attractor. Furthermore, under a quantitative separation condition on periodic orbits we show that the associated empirical periodic measures converge weakly to the normalized s_ℱ-dimensional Hausdorff measure on the self-similar attractor.

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BibTeX RIS

BibTeX

@misc{kong2026asymptotic,
  title = {{Asymptotic separation of periodic orbits and fractal dimension}},
  author = {Derong Kong and Zhiqiang Wang and Daohua Yu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16296},
  eprint = {2609.16296},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16296},
}

RIS

TY  - UNPB
TI  - Asymptotic separation of periodic orbits and fractal dimension
AU  - Derong Kong
AU  - Zhiqiang Wang
AU  - Daohua Yu
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.16296
UR  - https://arxiv.org/abs/2609.16296
AB  - For a totally bounded metric dynamical system $(X, d, T)$ we introduce a new critical value s(X, d, T) which quantifies the asymptotic separation of periodic orbits. More precisely, s(X,d,T) is defined to be the supremum of all s≥ 0 for which there exists a sequence of periodic orbits {𝒪_k}_k=1^∞ in (X,d,T) such that lim_k→ ∞ #𝒪_k = +∞ and liminf_k→∞#𝒪_k·η(𝒪_k)^s >0, where η(𝒪_k) denotes the smallest distance between distinct points in 𝒪_k. When the dynamical system is induced by a self-similar iterated function system ℱ satisfying the strong separation condition, we prove that this critical value is equal to the Hausdorff dimension s_ℱ of its self-similar attractor. Furthermore, under a quantitative separation condition on periodic orbits we show that the associated empirical periodic measures converge weakly to the normalized s_ℱ-dimensional Hausdorff measure on the self-similar attractor.
ER  -

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