胖 Sierpinski 垫片的唯一编码的临界底数
动力系统
2018-12-04 v1 组合数学
数论
摘要
给定 β ∈ ( 1 , 2 ) \beta\in(1,2) β ∈ ( 1 , 2 ) S β \mathcal S_\beta S β \[ f_{\beta,d}(x)=\frac{x+d}{\beta},\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\} \] 在 R 2 \mathbb R^2 R 2 S β \mathcal S_\beta S β P P P ( d i ) ∈ A N (d_i)\in\mathcal A^\mathbb N ( d i ) ∈ A N P = ∑ i = 1 ∞ d i / β i P=\sum_{i=1}^\infty d_i/\beta^i P = ∑ i = 1 ∞ d i / β i ( d i ) (d_i) ( d i ) P P P S β \mathcal S_\beta S β O β : = ⋃ c , d ∈ A , c ≠ d f β , c ( Δ β ) ∩ f β , d ( Δ β ) \mathcal O_\beta:=\bigcup_{c,d\in\mathcal A, c\ne d}f_{\beta,c}(\Delta_\beta)\cap f_{\beta,d}(\Delta_\beta) O β := ⋃ c , d ∈ A , c = d f β , c ( Δ β ) ∩ f β , d ( Δ β ) Δ β \Delta_\beta Δ β S β \mathcal S_\beta S β U ~ β : = { ∑ i = 1 ∞ d i β i ∈ S β : ∑ i = 1 ∞ d n + i β i ∉ O β ∀ n ≥ 0 } \widetilde{\mathcal U}_\beta:=\left\{\sum_{i=1}^\infty \frac{d_i}{\beta^i}\in \mathcal S_\beta: \sum_{i=1}^\infty\frac{d_{n+i}}{\beta^i}\notin\mathcal O_\beta~\forall n\ge 0\right\} U β := { ∑ i = 1 ∞ β i d i ∈ S β : ∑ i = 1 ∞ β i d n + i ∈ / O β ∀ n ≥ 0 } U ~ β \widetilde{\mathcal U}_\beta U β β c ≈ 1.55263 \beta_c\approx 1.55263 β c ≈ 1.55263 U ~ β \widetilde{\mathcal U}_\beta U β β > β c \beta>\beta_{c} β > β c β = β c \beta=\beta_c β = β c U ~ β \widetilde{\mathcal U}_\beta U β β < β c \beta<\beta_c β < β c U ~ β \widetilde{\mathcal U}_\beta U β U ~ β \widetilde{\mathcal U}_\beta U β
引用
@article{arxiv.1812.00585,
title = {Critical base for the unique codings of fat Sierpinski gasket},
author = {Derong Kong and Wenxia Li},
journal= {arXiv preprint arXiv:1812.00585},
year = {2018}
}
备注
28 pages, 10 figures