Open dynamics with a moving hole: inhomogeneous IFS and exponential separation condition
Abstract
Let ℱ={fi(x)=ri x+ti}i=0b−1 be a self-similar iterated function system (IFS) on ℝ with b≥ 3, and let Kℱ be the associated self-similar set. Given m∈ℕ, for any sequence ω=ω0ω1… with each ωi∈ {0,1,…,b−1}m a block of length m, let Kℱω:=πℱ({(di)∈ {0,1,…,b−1}ℕ: dk+1dk+2… dk+m≠ωk ∀ k≥ 0}), where πℱ is the natural coding map from {0,1,…,b−1}ℕ to Kℱ. The set Kℱω is called the survivor set of an open dynamical system with a moving hole related to ω. When ℱ is a homogeneous IFS satisfying the open set condition (OSC), the dimensions of Kℱω were determined by the first author, Sun and Wang (Adv. Math., 2026). In this paper we generalize these results to inhomogeneous IFSs. When ℱ satisfies the OSC, we give explicit formulae for the dimensions of Kℱω. In particular, we show that the packing dimension and upper box-counting dimension of Kℱω always coincide, while the Hausdorff dimension and lower box-counting dimension of Kℱω are not necessarily equal, which is different from the homogeneous case. Furthermore, when ℱ satisfies the exponential separation condition and the similarity dimension sℱ≤ 1, we are able to determine the Hausdorff dimension of Kℱω. Finally, we apply our results to badly approximable numbers and non-recurrence points in the setting of inhomogeneous IFS satisfying the OSC.
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BibTeX
@misc{kong2026open,
title = {{Open dynamics with a moving hole: inhomogeneous IFS and exponential separation condition}},
author = {Derong Kong and Kun Li and Cai-Yun Ma and Károly Simon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31128},
eprint = {2610.31128},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31128},
}
RIS
TY - UNPB
TI - Open dynamics with a moving hole: inhomogeneous IFS and exponential separation condition
AU - Derong Kong
AU - Kun Li
AU - Cai-Yun Ma
AU - Károly Simon
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31128
UR - https://arxiv.org/abs/2610.31128
AB - Let ℱ={f_i(x)=r_i x+t_i}_i=0^b−1 be a self-similar iterated function system (IFS) on ℝ with b≥ 3, and let K_ℱ be the associated self-similar set. Given m∈ℕ, for any sequence ω=ω^0ω^1… with each ω^i∈ {0,1,…,b−1}^m a block of length m, let K_ℱ^ω:=π_ℱ({(d_i)∈ {0,1,…,b−1}^ℕ: d_k+1d_k+2… d_k+m≠ω^k ∀ k≥ 0}), where π_ℱ is the natural coding map from {0,1,…,b−1}^ℕ to K_ℱ. The set K_ℱ^ω is called the survivor set of an open dynamical system with a moving hole related to ω. When ℱ is a homogeneous IFS satisfying the open set condition (OSC), the dimensions of K_ℱ^ω were determined by the first author, Sun and Wang (Adv. Math., 2026). In this paper we generalize these results to inhomogeneous IFSs. When ℱ satisfies the OSC, we give explicit formulae for the dimensions of K_ℱ^ω. In particular, we show that the packing dimension and upper box-counting dimension of K_ℱ^ω always coincide, while the Hausdorff dimension and lower box-counting dimension of K_ℱ^ω are not necessarily equal, which is different from the homogeneous case. Furthermore, when ℱ satisfies the exponential separation condition and the similarity dimension s_ℱ≤ 1, we are able to determine the Hausdorff dimension of K_ℱ^ω. Finally, we apply our results to badly approximable numbers and non-recurrence points in the setting of inhomogeneous IFS satisfying the OSC.
ER -