| 陈阳洋,赵云.一列连续函数的遍历优化[J].数学年刊A辑,2013,34(5):589~598 |
| 一列连续函数的遍历优化 |
| Ergodic Optimization for a Sequence of Continuous Functions |
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| DOI： |
| 中文关键词: 遍历测度, 次可加势函数, 最大化测度 |
| 英文关键词:Ergodic measures, Subadditive potentials, Maximizing measures |
| 基金项目:国家自然科学基金 (No.11001191), 教育部博士点基金 (No.20103201120001)和大学生创新性实验计划 (No.111028508) |
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| 中文摘要: |
| 设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq
1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in
\mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac
1 n f_n (x)=\sup\limits_{x\in X}
\limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x)
=\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n
(x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$
表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int
f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了
一些相关的应用. |
| 英文摘要: |
| Let $T:X\rightarrow X$ be a continuous
map on a compact metric space $X$, and $\mathcal{F}=\{f_n\}_{n\geq
1}$ a sequence of continuous functions on $X$. If $\mathcal{F}$
is asymptotically subadditive, then $\sup\limits_{x\in
\mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac
1 n f_n (x)=\sup\limits_{x\in X}
\limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x)
=\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n
(x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, where
$\mathcal{M}_T$ denotes the space of $T$-invariant Borel
probability measures, $\mathrm{Reg}(\mathcal{F},T)$ denotes the
set of all regular points for $\mathcal{F}$, and
$\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int
f_n \mathrm{d}\mu$. This generalizes some results of
Jenkinson, Schreiber, Sturman etc to asymptotically subadditive potentials. A sufficient condition for the subordination
principle of a subadditive potential is also provided. Some applications are given
at the end of this paper. |
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