Irregular sets, the $\beta$-transformation and the almost specification property
Irregular sets, the $\beta$-transformation and the almost specification property
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发表时间：
2009-05
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通讯作者：
D. Thompson
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作者：
D. Thompson
设$（X，d）$是一个紧度量空间，$f：X \mapsto X$是一个连续映射，满足一个我们称之为几乎规格的性质（比Pfister和Sullivan的$g$-几乎乘积性质稍弱），$\phi$是X$上的一个连续函数。我们表明，一组点的Birkhoff平均$\phi$不存在（我们称之为不规则集）要么是空的或有完整的拓扑熵。每一个$\beta$-移位满足几乎规范，我们证明了任何$\beta$-移位或$\beta$-变换的不规则集要么是空的，要么是具有全拓扑熵和Hausdorff维数的.
Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\phi$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $\phi$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $\beta$-shift satisfies almost specification and we show that the irregular set for any $\beta$-shift or $\beta$-transformation is either empty or has full topological entropy and Hausdorff dimension.