﻿ 渐近可加势的“历史集”的Hausdorff维数谱 Spectrum of Hausdorff Dimension on the Historic Set of the Asymptotically Additive Potentials

Pure Mathematics
Vol. 08  No. 06 ( 2018 ), Article ID: 27719 , 11 pages
10.12677/PM.2018.86093

Spectrum of Hausdorff Dimension on the Historic Set of the Asymptotically Additive Potentials

Tonghui Peng, Yalin Wang, Lingfang Xu, Guanzhong Ma

School of Mathematics and Statistics, Anyang Normal University, Anyang Henan

Received: Nov. 2nd, 2018; accepted: Nov. 13th, 2018; published: Nov. 26th, 2018

ABSTRACT

Authors conduct multifractal analysis of historic set of the asymptotically additive potentials on a class of non-uniformly expanding systems. They prove that either the historic set is empty or carries full Hausdorff dimension.

Keywords:Non-Uniformly Expanding, Asymptotically Additive Potentials, Historic Set

1. 引言

1.1. “历史集”

$\left(X\text{,}T\right)$ 为拓扑动力系统，即X为紧度量空间， $T:X\to X$ 为连续自映射， $C\left(X,ℝ\right)$ 表示从X到 $ℝ$ 的连续函数全体。给定 $\varphi \in C\left(X,ℝ\right)$ ，称 ${S}_{n}\varphi :=\underset{i=0}{\overset{n-1}{\sum }}\varphi \circ {T}^{i}$${A}_{n}\varphi :=\frac{1}{n}{S}_{n}\varphi$ 分别为 $\varphi$ 的Birkhoff和与Birkhoff均值。若存在 $\varphi \in C\left(X,R\right)$ ，使得极限 $\underset{n\to \infty }{\mathrm{lim}}{A}_{n}\varphi \left(x\right)$ 不存在，则称点x的轨道

$\left\{x,Tx,{T}^{2}x,\cdots \right\}$

1.2. 系统与符号

1、若 $i\ne j$ ，则 $\mathrm{int}\left({I}_{i}\right)\cap \mathrm{int}\left({I}_{j}\right)=\varnothing$ ，本文中 $\mathrm{int}\left({I}_{i}\right)$ 指集合 ${I}_{i}$ 的内部；

2、对 $1\le i\le m$$T{|}_{{I}_{i}}:{I}_{i}\to \left[0,1\right]$${C}^{1+r}$ 的满射，且存在唯一 ${x}_{i}\in {I}_{i}$ 满足 $T\left({x}_{i}\right)={x}_{i}$${T}^{\prime }\left({x}_{i}\right)\ge 1$。若 ${T}^{\prime }\left({x}_{i}\right)=1$ ，则称 ${x}_{i}$ 为抛物不动点，否则称 ${x}_{i}$ 为扩张不动点；

3、若 $x\notin \left\{{x}_{1},{x}_{2},\cdots ,{x}_{m}\right\}$ ，则 ${T}^{\prime }\left(x\right)>1$

$\Lambda :=\left\{x\in {\cup }_{j=1}^{m}{I}_{j}|{T}^{n}\left(x\right)\in I,对一切n\ge 0\right\}$.

$\Pi \left(\omega \right):=\underset{n\to \infty }{\mathrm{lim}}{T}_{{\omega }_{1}}\circ {T}_{{\omega }_{2}}\cdots \circ {T}_{{\omega }_{n}}\left(\left[0,1\right]\right)$.

1.3. 渐近可加势

$\Phi ={\left({\varphi }_{n}\right)}_{n=1}^{\infty }$ 为一列定义在X上的连续函数，若对任意给定的 $\epsilon >0$ ，都存在 ${g}_{\epsilon }\in C\left(X,ℝ\right)$ ，使得

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{n}‖{\varphi }_{n}-{S}_{n}{g}_{\epsilon }‖<\epsilon$,

$\Phi$ 为渐近可加势，其中对给定的 $f\in C\left(X,ℝ\right)$$‖f‖$ 是上确界范数。

$\Phi$ 满足下述条件，称 $\Phi$ 为可加势，对 $\forall n\ge 1$$\forall m\ge 1$$\forall x\in X$ ，都有 ${\varphi }_{n+m}\left(x\right)={\varphi }_{n}\left(x\right)+{\varphi }_{m}\left({T}^{n}x\right)$

1、对每个 $n\ge 1$${\varphi }_{n}:X\to ℝ$ 连续；

2、存在正常数 $C\left(\Phi \right)$ ，对任意给定的 $n,p\in ℕ$ 和任意给定的 $x\in X$ ，有下式成立，

$|{\varphi }_{n+p}\left(x\right)-{\varphi }_{n}\left(x\right)-{\varphi }_{p}\left({T}^{n}x\right)|\le C\left(\Phi \right)$.

1.4. 主要结果

1、 $\text{H}\left(\Phi \right)=\varnothing$ 当且仅当 ${\Omega }_{\Phi }$ 是单点集；

2、 $\text{H}\left(\Phi \right)\ne \varnothing$ 当且仅当 ${\mathrm{dim}}_{H}\text{H}\left(\Phi \right)={\mathrm{dim}}_{H}\Lambda$

2. 记号和预备知识

$A=\left\{1,2,\cdots ,m\right\}$$\Sigma ={A}^{ℕ}$ ，令 ${\Sigma }_{n}:=\left\{w={w}_{1}\cdots {w}_{n}|{w}_{i}\in A\right\}$ 是长为n的词的全体。 $\omega ={\left\{{\omega }_{n}\right\}}_{n=1}^{\infty }\in \Sigma$ ，记 $\omega {|}_{n}={\omega }_{1}\cdots {\omega }_{n}$。对词 $w\in {\Sigma }_{n}$ ，令 $\left[w\right]:=\left\{\omega \in \Sigma |\omega {|}_{n}=w\right\}$ 是由w确定的长为n的柱集。对给定的连续函数 $\varphi :\Sigma \to ℝ$ ，称

${\text{Var}}_{n}\varphi :=\underset{\omega {|}_{n}=\tau {|}_{n}}{\mathrm{sup}}|{\varphi }_{n}\left(\omega \right)-{\varphi }_{n}\left(\tau \right)|$

$\varphi$ 的n-级变差，同时令 $‖\varphi ‖:=\underset{\tau \in \Pi }{\mathrm{sup}}|\varphi \left(\tau \right)|$

$\stackrel{^}{\Lambda }:=\left\{x\in \Lambda |#\left\{{\Pi }^{-1}\left(x\right)\right\}=2\right\}$ 表示吸引子中有两个编码的点，其中 $#A$ 表示集合A的元素个数。在本文条件下， $\stackrel{^}{\Lambda }$${\Pi }^{-1}\stackrel{^}{\Lambda }$ 都是至多可数集， $\Pi :\Sigma \{\Pi }^{-1}\stackrel{^}{\Lambda }\to \Lambda \\stackrel{^}{\Lambda }$ 是双射。对长为n的词 $w={w}_{1}{w}_{2}\cdots {w}_{n}$ ，记 ${I}_{w}={T}_{{w}_{1}}\circ {T}_{{w}_{2}}\circ \cdots \circ {T}_{{w}_{n}}\left(I\right)$ ，对 $\omega \in \Sigma$ ，记 ${I}_{n}\left(\omega \right)={I}_{\omega {|}_{n}}$。用 ${D}_{n}\left(\omega \right)$ 表示 ${I}_{n}\left(\omega \right)$ 的直径，同时令 ${g}_{n}\left(\omega \right):=-\mathrm{log}{{T}^{\prime }}_{{\omega }_{1}}\Pi \left(\sigma \omega \right)$ ，令

${A}_{n}g\left(\omega \right):=\frac{1}{n}\underset{i=0}{\overset{n-1}{\sum }}g\left({T}^{i}\left(\omega \right)\right)$.

$\underset{n\to \infty }{\mathrm{lim}}\underset{\omega \in \Sigma }{\mathrm{sup}}\left\{|-\frac{1}{n}\mathrm{log}{D}_{n}\left(\omega \right)-{A}_{n}g\left(\omega \right)|\right\}=0$.

${\stackrel{^}{\lambda }}_{n}\left(\omega \right)=-\frac{1}{n}\mathrm{log}{D}_{n}\left(\omega \right)$ ，对给定的σ-不变测度 $\mu$ ，令 $\lambda \left(\mu ,\sigma \right):={\int }_{\Sigma }g\text{d}\mu$$\mu$ 的Lyapunov指数， ${\Pi }_{*}\mu :=\mu \circ {\Pi }^{-1}$$\mu$ 的像测度。下面引理组合了文献 [18] 中的引理2和引理3，在定理2.1的证明中起着本质作用。

$h\left({\mu }_{n},\sigma \right)\to h\left(\mu ,\sigma \right)$, $\lambda \left({\mu }_{n},\sigma \right)\to \lambda \left(\mu ,\sigma \right)$.

$\underset{n\to \infty }{\mathrm{lim}}\underset{m\to \infty }{\mathrm{lim}}\frac{1}{m}\int {\varphi }_{m}\text{d}{\mu }_{n}=\underset{m\to \infty }{\mathrm{lim}}\frac{1}{m}\int {\varphi }_{m}\text{d}\mu$.

$\underset{n\to \infty }{\mathrm{lim}}{\Phi }_{*}\left({\mu }_{n}\right)={\Phi }_{*}\left(\mu \right)$

3. 定理2.1的证明

$\frac{1}{n}|{\varphi }_{n}\left(x\right)-{S}_{n}h\left(x\right)|<\epsilon$,

$\stackrel{¯}{\alpha }\le {\int }_{X}h\text{d}\mu +\epsilon \le \underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}{\int }_{X}{\varphi }_{n}\text{d}\mu +2\epsilon ={\Phi }_{*}\left(\mu \right)+2\epsilon$.

${\Phi }_{*}\left(\nu \right)\le \underset{_}{\alpha }+2\epsilon <\stackrel{¯}{\alpha }-2\epsilon \le {\Phi }_{*}\left(\mu \right)$ ，即 ${\Phi }_{*}\left(\nu \right)\ne {\Phi }_{*}\left(\mu \right)$.

$\stackrel{^}{\Phi }=\Phi \circ \Pi$ ，易证 ${\Omega }_{\Phi }={\Omega }_{\stackrel{^}{\Phi }}$$Η\left(\Phi \right)=\Pi Η\left(\stackrel{^}{\Phi }\right)$。则定理1是下述引理的直接推论。

${\mathrm{dim}}_{\text{H}}Η\left(\Phi \right)\ge \mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{h\left(\nu ,\sigma \right)}{\lambda \left(\nu ,\sigma \right)}\right\}$.

${\mathrm{dim}}_{\text{H}}\Lambda =\underset{\mu \in M\left(\Sigma ,\sigma \right)}{\mathrm{sup}}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)}\text{|}\lambda \left(\mu ,\sigma \right)>0\right\}$.

$\begin{array}{l}{\mathrm{dim}}_{\text{H}}Η\left(\Phi \right)\ge \mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{h\left({\mu }_{s},\sigma \right)}{\lambda \left({\mu }_{s},\sigma \right)}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{sh\left(\mu ,\sigma \right)+\left(1-s\right)h\left(\nu ,\sigma \right)}{s\lambda \left(\mu ,\sigma \right)+\left(1-s\right)\lambda \left(\nu ,\sigma \right)}\right\}\end{array}$.

$s\to 1$ ，可得由 $\epsilon$ 的任意性，定理1得证。下面证明引理4。

${\mathrm{dim}}_{\text{H}}\Pi M\ge \mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{h\left(\nu ,\sigma \right)}{\lambda \left(\nu ,\sigma \right)}\right\}$.

$\left\{\begin{array}{l}{\mathrm{var}}_{n}{A}_{n}g<{\epsilon }_{2i-1},\\ \underset{\omega \in \Sigma }{\mathrm{max}}|{\stackrel{^}{\lambda }}_{n}\left(\omega \right)-{A}_{n}g\left(\omega \right)|<{\epsilon }_{2i-1}.\end{array}$ (1)

$\left\{\begin{array}{l}|h\left({\mu }_{2i-1},\sigma \right)-h\left(\mu ,\sigma \right)|<{\epsilon }_{2i-1},\\ |{\stackrel{^}{\Phi }}_{*}\left({\mu }_{2i-1}\right)-\alpha |<{\epsilon }_{2i-1},\\ |\lambda \left({\mu }_{2i-1},\sigma \right)-\lambda \left(\mu ,\sigma \right)|<{\epsilon }_{2i-1}.\end{array}$ (2)

${\alpha }_{2i-1}={\stackrel{^}{\Phi }}_{*}\left({\mu }_{2i-1}\right)$ ，注意到 ${\mu }_{2i-1}$ 的遍历性，由Birkhoff遍历定理和Shanon-Mcmillan-Breiman定理，对 ${\mu }_{2i-1}$ a。e。的 $\omega \in \Sigma$ ，有下式成立，

$\left\{\begin{array}{l}\frac{{\stackrel{^}{\varphi }}_{n}\left(\omega \right)}{n}\to \alpha ,\\ {A}_{n}g\left(\omega \right)\to \lambda \left({\mu }_{2i-1},\sigma \right),\\ -\frac{\mathrm{log}{\mu }_{2i-1}\left(\left[\omega {|}_{n}\right]\right)}{n}\to h\left({\mu }_{2i-1},\sigma \right).\end{array}$ (3)

$\left\{\begin{array}{l}|\frac{1}{n}{\stackrel{^}{\varphi }}_{n}\left(\omega \right)-{\alpha }_{2i-1}|<{\epsilon }_{2i-1},\\ |{A}_{n}g\left(\omega \right)-\lambda \left({\mu }_{2i-1},\sigma \right)|<{\epsilon }_{2i-1},\\ |-\frac{\mathrm{log}{\mu }_{2i-1}\left(\left[\omega {|}_{n}\right]\right)}{n}-h\left({\mu }_{2i-1},\sigma \right)|<{\epsilon }_{2i-1}.\end{array}$ (4)

$\Sigma \left(2i-1\right)=\left\{\omega {|}_{{l}_{2i-1}}|\omega \in {\Omega }^{\prime }\left(2i-1\right)\right\}$ ，称  为(2i-1)-级Moran块， $\Sigma \left(2i-1\right)$ 是构造Moran集的基本模块。再令

$\Omega \left(2i-1\right)=\underset{\omega \in \Sigma \left(2i-1\right)}{\cup }\left[\omega \right]$,

$\begin{array}{l}{\mu }_{2i-1}\left(\Omega \left(2i-1\right)\right)\ge {\mu }_{2i-1}\left({\Omega }^{\prime }\left(2i-1\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\ge 1-\delta \end{array}$.

$\left\{\begin{array}{l}{\mathrm{var}}_{n}{A}_{n}g<{\epsilon }_{2i},\\ \underset{\omega \in \Sigma }{\mathrm{max}}|{\stackrel{^}{\lambda }}_{n}\left(\omega \right)-{A}_{n}g\left(\omega \right)|<{\epsilon }_{2i}.\end{array}$ (5)

$\left\{\begin{array}{l}|h\left({\nu }_{2i},\sigma \right)-h\left(\nu ,\sigma \right)|<{\epsilon }_{2i},\\ |{\stackrel{^}{\Phi }}_{*}\left({\nu }_{2i}\right)-\beta |<{\epsilon }_{2i},\\ |\lambda \left({\nu }_{2i},\sigma \right)-\lambda \left(\nu ,\sigma \right)|<{\epsilon }_{2i}.\end{array}$ (6)

${\stackrel{^}{\Phi }}_{\text{*}}\left({\nu }_{2i}\right)={\beta }_{2i}$ ，由 ${\nu }_{2i}$ 的遍历性可知，对 ${\nu }_{2i}$ a..e.的 $\omega \in \Sigma$ 有下列结果，

$\left\{\begin{array}{l}\frac{{\stackrel{^}{\varphi }}_{n}\left(\omega \right)}{n}\to {\beta }_{2i},\\ {A}_{n}g\left(\omega \right)\to \lambda \left({\nu }_{2i},\sigma \right),\\ -\frac{\mathrm{log}{\nu }_{2i}\left(\left[\omega {|}_{n}\right]\right)}{n}\to h\left({\nu }_{2i},\sigma \right).\end{array}$ (7)

$\left\{\begin{array}{l}|\frac{1}{n}{\stackrel{^}{\varphi }}_{n}\left(\omega \right)-{\beta }_{2i}|<{\epsilon }_{2i},\\ |{A}_{n}g\left(\omega \right)-\lambda \left({\nu }_{2i},\sigma \right)|<{\epsilon }_{2i},\\ |-\frac{\mathrm{log}{\nu }_{2i}\left(\left[\omega {|}_{n}\right]\right)}{n}-h\left({\nu }_{2i},\sigma \right)|<{\epsilon }_{2i}.\end{array}$ (8)

$\begin{array}{l}\Sigma \left(2i\right)=\left\{\omega {|}_{{l}_{2i}}|\omega \in {\Omega }^{\prime }\left(2i\right)\right\},\\ \Omega \left(2i\right)=\underset{w\in \Sigma \left(2i\right)}{\cup }\left[w\right]\end{array}$

${N}_{0}=1$ ，对 $i\ge 1$ ，令 ${N}_{i}={2}^{{l}_{i+2}+{N}_{i-1}}$${N}_{i}$ 是在第i层把Moran块重复拼砌的次数。令Moran集M为下述集合，

$\stackrel{{}_{{N}_{1}}}{\stackrel{︹}{\Sigma \left(1\right)\cdots \Sigma \left(1\right)}}\cdots \stackrel{{N}_{i}}{\stackrel{︹}{\Sigma \left(i\right)\cdots \Sigma \left(i\right)}}\cdots$.

${N}_{i}$ 的选择在估计Moran集的Hausdorff维数时是至关重要的。

$j\ge \text{1}$ ，令 ${n}_{j}=\underset{i=1}{\overset{j}{\sum }}{l}_{i}{N}_{i}$ ，固定 $\omega \in M$ ，注意到

$\underset{j\to \infty }{\mathrm{lim}}\frac{{l}_{2}{N}_{2}+{l}_{4}{N}_{4}+\cdots +{l}_{2j}{N}_{2j}}{{n}_{2j+1}}=0$.

$\begin{array}{l}\underset{j\to \infty }{\mathrm{lim}}\frac{{\stackrel{^}{\varphi }}_{{n}_{2j+1}}\left(\omega \right)}{{n}_{2j+1}}=\alpha ,\\ \underset{j\to \infty }{\mathrm{lim}}\frac{{\stackrel{^}{\varphi }}_{{n}_{2j}}\left(\omega \right)}{{n}_{2j}}=\beta \end{array}$

$\stackrel{{}_{{N}_{1}}}{\stackrel{︹}{{l}_{1}\cdots {l}_{1}}}\cdots \stackrel{{N}_{i}}{\stackrel{︹}{{l}_{i}\cdots {l}_{i}}}\cdots$

$\stackrel{{}_{{N}_{1}}}{\stackrel{︹}{\Sigma \left(1\right)\cdots \Sigma \left(1\right)}}\cdots \stackrel{{N}_{i}}{\stackrel{︹}{\Sigma \left(i\right)\cdots \Sigma \left(i\right)}}\cdots$

$\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\le n<\underset{i=1}{\overset{J\left(n\right)+1}{\sum }}{l}_{i}^{*}$ ;

$\underset{i=1}{\overset{r\left(n\right)}{\sum }}{N}_{i}\le J\left(n\right)<\underset{i=1}{\overset{r\left(n\right)+1}{\sum }}{N}_{i}$.

$J\left(n\right)\le J\left(n+1\right)\le J\left(n\right)+1,{l}_{J\left(n\right)+1}^{*}={l}_{r\left(n\right)+1}且{l}_{J\left(n\right)+2}^{*}\le {l}_{r\left(n\right)+2}$ 9)

$j=1,2$ ，有

$\frac{{l}_{J\left(n\right)+j}^{*}}{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}}\le \frac{{l}_{r\left(n\right)+j}}{{N}_{r\left(n\right)}{l}_{r\left(n\right)}}$,

$\frac{{l}_{J\left(n\right)+j}^{*}}{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}}\to 0,且\frac{\underset{i=1}{\overset{J\left(n\right)+1}{\sum }}{l}_{i}^{*}}{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}}\to 1$ (10)

${\rho }_{\omega }^{i}=\frac{{\eta }_{i}^{*}\left[\omega \right]}{{\eta }_{i}^{*}\left({\Omega }^{*}\left(i\right)\right)}$.

$\stackrel{^}{\eta }\left(\left[w\right]\right):=\underset{i=1}{\overset{n}{\prod }}{\rho }_{{w}_{i}}^{i}$,

$\underset{r↓0}{\mathrm{lim}\mathrm{inf}}\frac{\mathrm{log}{\Pi }_{*}\eta B\left(x,r\right)}{\mathrm{log}r}\ge \mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{h\left(\nu ,\sigma \right)}{\lambda \left(\nu ,\sigma \right)}\right\}$ (11)

${\mathrm{dim}}_{\text{H}}{\Pi }_{*}\eta :=\mathrm{sup}\left\{s\ge 0|\underset{r↓0}{\mathrm{lim}\mathrm{inf}}\frac{\mathrm{log}{\Pi }_{*}\eta B\left(x,r\right)}{\mathrm{log}r}\ge s对{\Pi }_{*}\eta \text{a}\text{.e}\text{.}x\in \Lambda \right\}$

${\mathrm{dim}}_{\text{H}}Η\left(\Phi \right)\ge {\mathrm{dim}}_{\text{H}}\Pi M\ge {\mathrm{dim}}_{\text{H}}{\Pi }_{*}\eta \ge \mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{h\left(\nu ,\sigma \right)}{\lambda \left(\nu ,\sigma \right)}\right\}$.

$n{\stackrel{^}{\lambda }}_{n}\left(\omega \right)\le \underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(\lambda \left({\tau }_{i},\sigma \right)+4{\epsilon }_{i}^{*}\right)+{l}_{J\left(n\right)+1}^{*}\left(‖g‖+{\epsilon }_{J\left(n\right)}^{*}\right):=\rho \left(n\right)$.

${\text{e}}^{-\rho \left(n+1\right)}\le r<{\text{e}}^{-\rho \left(n\right)}$ (12)

$C:=\left\{{I}_{n}\left(\omega \right)|\omega \in M且{I}_{n}\left(\omega \right)\cap B\left(x,r\right)\ne \varnothing \right\}$.

${D}_{n}\left(\omega \right)\ge {\text{e}}^{-\rho \left(n\right)}$ 可知C至多含有3个元素。任取 $\omega \in M$ 满足 ${I}_{n}\left(\omega \right)\in C$ ，则 $\omega {|}_{n}={w}_{1}{w}_{2}\cdots {w}_{J\left(n\right)}v$ ，其中，v是 ${\Sigma }^{*}\left(J\left(n\right)+1\right)$ 中元素 $\stackrel{^}{v}$ 的前缀，则有

$\begin{array}{c}{\Pi }_{*}\eta \left({I}_{n}\left(\omega \right)\right)=\underset{i=1}{\overset{J\left(n\right)}{\prod }}\frac{{\eta }_{i}^{*}\left(\left[{w}_{i}\right]\right)}{{\eta }_{i}^{*}\left({\Omega }^{*}\left(i\right)\right)}\cdot \frac{{\eta }_{J\left(n\right)+1}^{*}\left(\left[v\right]\right)}{{\eta }_{J\left(n\right)+1}^{*}\left({\Omega }^{*}\left(J\left(n\right)+1\right)\right)}\\ \le {\left(1-\delta \right)}^{-J\left(n\right)-1}\underset{i=1}{\overset{J\left(n\right)}{\prod }}{\eta }_{i}^{*}\left(\left[{w}_{i}\right]\right)\end{array}$.

$\begin{array}{c}\mathrm{log}{\Pi }_{*}\eta \left(B\left(x,r\right)\right)\le -\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(-\frac{\mathrm{log}{\eta }_{i}^{*}\left(\left[{w}_{i}\right]\right)}{{l}_{i}^{*}}\right)-\left(J\left(n\right)+1\right)\mathrm{log}\left(1-\delta \right)+\mathrm{log}3\\ \le -\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(h\left({\tau }_{i},\sigma \right)-2{\epsilon }_{i}^{*}\right)-\left(J\left(n\right)+1\right)\mathrm{log}\left(1-\delta \right)+\mathrm{log}3\end{array}$.

$\begin{array}{c}\underset{r↓0}{\mathrm{lim}\mathrm{inf}}\frac{\mathrm{log}{\Pi }_{*}\eta \left(B\left(x,r\right)\right)}{\mathrm{log}r}\ge \underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(h\left({\tau }_{i},\sigma \right)-2{\epsilon }_{i}^{*}\right)+\left(J\left(n\right)+1\right)\mathrm{log}\left(1-\delta \right)-\mathrm{log}3}{\underset{i=1}{\overset{J\left(n+1\right)}{\sum }}{l}_{i}^{*}\left(\lambda \left({\tau }_{i},\sigma \right)+4{\epsilon }_{i}^{*}\right)+{l}_{J\left(n+1\right)+1}^{*}\left(‖g‖+{\epsilon }_{J\left(n+1\right)+1}^{*}\right)}\\ =\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(h\left({\tau }_{i},\sigma \right)-2{\epsilon }_{i}^{*}\right)}{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(\lambda \left({\tau }_{i},\sigma \right)+4{\epsilon }_{i}^{*}\right)}\\ \ge \underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(\lambda \left({\tau }_{i},\sigma \right)+4{\epsilon }_{i}^{*}\right)\mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)-2{\epsilon }_{i}^{*}}{\lambda \left(\mu ,\sigma \right)+4{\epsilon }_{i}^{*}},\frac{h\left(\nu ,\sigma \right)-2{\epsilon }_{i}^{*}}{\lambda \left(\nu ,\sigma \right)+4{\epsilon }_{i}^{*}}\right\}}{\underset{i=1}{\overset{J\left(n\right)}{\sum }}{l}_{i}^{*}\left(\lambda \left({\tau }_{i},\sigma \right)+4{\epsilon }_{i}^{*}\right)}\\ \ge \mathrm{min}\left\{\frac{h\left(\mu ,\sigma \right)}{\lambda \left(\mu ,\sigma \right)},\frac{h\left(\nu ,\sigma \right)}{\lambda \left(\nu ,\sigma \right)}\right\}\end{array}$

${\left({a}_{n}\right)}_{n=1}^{\infty }$${\left({b}_{n}\right)}_{n=1}^{\infty }$ 和是两个正实数序列，则有

$\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\underset{k=1}{\overset{n}{\sum }}{a}_{k}}{\underset{k=1}{\overset{n}{\sum }}{b}_{k}}\ge \underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{{a}_{n}}{{b}_{n}}$.

(11)式得证。

Spectrum of Hausdorff Dimension on the Historic Set of the Asymptotically Additive Potentials[J]. 理论数学, 2018, 08(06): 688-698. https://doi.org/10.12677/PM.2018.86093

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