论文标题
一些马尔可夫型措施的量级错误的渐近学渐近误差具有完整的重叠
Asymptotics of the quantization errors for some Markov-type measures with complete overlaps
论文作者
论文摘要
令$ \ mathcal {g} $为有针对性的图形，顶点$ 1,2，\ ldots，2n $。令$ \ mathcal {t} =（t_ {i，j}）_ {（i，j）\ in \ mathcal {g}} $是一个承诺相似的家族。对于$ 1 \ leq i \ leq n $，让$ i^+：= i+n $。对于$ 1 \ leq i，j \ leq n $，我们定义$ \ Mathcal {m} _ {i，j} = \ {（i，j），（i，j^+），（i^+，j），（i^+，j^+，j^+）我们假设每个$（\ widetilde {\ widetilde {i}，\ wideTilde {i}，\ wideTilde {j} =令$ k $表示由$ \ Mathcal {t} $确定的Mauldin-Williams分形。令$χ=（χ_i）_ {i = 1}^{2n} $为正概率向量，而$ p $ a行 - 策略矩阵，可作为$ \ Mathcal {g} $的入击矩阵。我们用$ν$表示与$χ$和$ p $相关的马尔可夫型措施。令$ω= \ {1，\ ldots，2n \} $和$ g_ \ infty = \ {σ\inΩ^{\ mathbb {n}} :(σ_i，σ_i，σ_{i+1}）令$π$为自然投影，从$ g_ \ infty $到$ k $，$μ=ν\circπ^{ - 1} $。我们考虑以下两种情况：1。$ \ Mathcal {G} $具有两个由$ N $ VERTICES组成的紧密连接组件； 2。$ \ MATHCAL {G} $连接。对于$ \ Mathcal {g} $和$ \ Mathcal {t} $的一些假设，对于情况1，我们确定量子$ d_r（μ）$ $ $ $ $ $的确切值$ s_r $，并证明$ s_r $ s_r $ dimential-dimentional-dimentional-dimentional-dimentional-dimentiation soughtiation s量化系数始终是肯定的，但是ump opter umper the ump ins nisite;我们为$μ$有限的高量化系数建立了必要的条件；对于情况2，我们在类似压力的函数方面确定$ d_r（μ）$，并证明$ d_r（μ）$ - 尺寸上限和下量化系数均为正和有限。
Let $\mathcal{G}$ be a directed graph with vertices $1,2,\ldots, 2N$. Let $\mathcal{T}=(T_{i,j})_{(i,j)\in\mathcal{G}}$ be a family of contractive similitudes. For every $1\leq i\leq N$, let $i^+:=i+N$. For $1\leq i,j\leq N$, we define $\mathcal{M}_{i,j}=\{(i,j),(i,j^+),(i^+,j),(i^+,j^+)\}\cap\mathcal{G}$. We assume that $T_{\widetilde{i},\widetilde{j}}=T_{i,j}$ for every $(\widetilde{i},\widetilde{j})\in \mathcal{M}_{i,j}$. Let $K$ denote the Mauldin-Williams fractal determined by $\mathcal{T}$. Let $χ=(χ_i)_{i=1}^{2N}$ be a positive probability vector and $P$ a row-stochastic matrix which serves as an incidence matrix for $\mathcal{G}$. We denote by $ν$ the Markov-type measure associated with $χ$ and $P$. Let $Ω=\{1,\ldots,2N\}$ and $G_\infty=\{σ\inΩ^{\mathbb{N}}:(σ_i,σ_{i+1})\in\mathcal{G}, \;i\geq 1\}$. Let $π$ be the natural projection from $G_\infty$ to $K$ and $μ=ν\circπ^{-1}$. We consider the following two cases: 1. $\mathcal{G}$ has two strongly connected components consisting of $N$ vertices; 2. $\mathcal{G}$ is strongly connected. With some assumptions for $\mathcal{G}$ and $\mathcal{T}$, for case 1, we determine the exact value $s_r$ of the quantization dimension $D_r(μ)$ for $μ$ and prove that the $s_r$-dimensional lower quantization coefficient is always positive, but the upper one can be infinite; we establish a necessary and sufficient condition for the upper quantization coefficient for $μ$ to be finite; for case 2, we determine $D_r(μ)$ in terms of a pressure-like function and prove that $D_r(μ)$-dimensional upper and lower quantization coefficient are both positive and finite.
