论文标题
部分可观测时空混沌系统的无模型预测
Projections of four corner Cantor set: total self-similarity, spectrum and unique codings
论文作者
论文摘要
给定的$ρ\在（0,1/4] $中，四个角Cantor集$ e \ subset \ mathbb {r}^{2} $是由迭代功能系统生成的自相似集 \ [ \ left \ {（ρx，ρy），\ quad（ρx，ρy+1-ρ），\ quad（ρx+1-ρ，ρy），\ quad（ρx+1-ρ，ρy+1- p）\ right \}。 \] 对于$θ\在[0，π）$中，让$e_θ$是$ e $的正交投影到一条线上，$ x $轴的角度$θ$。在本文中，我们给出了一个完整的表征，其中投影$e_θ$完全相似。我们还研究了$e_θ$的频谱，事实证明，$e_θ$的频谱在且仅当$e_θ$完全相似时才能达到其最大值。此外，当$e_θ$完全相似时，我们计算其Hausdorff尺寸并研究子集$u_θ$，该子集由E_θ$中的所有$ x \组成，具有独特的编码。特别是，我们表明$ \ dim_hu_θ= \ dim_he_θ$对于[0，π）$中的几乎每个$θ\。最后，对于$ρ= 1/4 $，我们描述了$e_θ$包含一个间隔的$θ$的分布。事实证明，$e_θ$包含一个间隔的可能性小于$e_θ$具有精确重叠的可能性。
Given $ρ\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system \[ \left\{(ρx, ρy), \quad(ρx, ρy+1-ρ),\quad (ρx+1-ρ, ρy),\quad(ρx+1-ρ,ρy+1-ρ)\right\}. \] For $θ\in[0,π)$ let $E_θ$ be the orthogonal projection of $E$ onto a line with an angle $θ$ to the $x$-axis. In this paper we give a complete characterization on which the projection $E_θ$ is totally self-similar. We also study the spectrum of $E_θ$, which turns out that the spectrum of $E_θ$ achieves its maximum value if and only if $E_θ$ is totally self-similar. Furthermore, when $E_θ$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_θ$ which consists of all $x\in E_θ$ having a unique coding. In particular, we show that $\dim_H U_θ=\dim_H E_θ$ for Lebesgue almost every $θ\in[0,π)$. Finally, for $ρ=1/4$ we describe the distribution of $θ$ in which $E_θ$ contains an interval. It turns out that the possibility for $E_θ$ to contain an interval is smaller than that for $E_θ$ to have an exact overlap.
