关于连分数的Khintchine指数和李雅普诺夫指数
On Khintchine exponents and Lyapunov exponents of continued fractions
摘要：
假设$x\in [0,1) $允许其连分数展开$x=[a_1(x), a_2(x),...]$。当极限存在时，$x$的Khintchine指数$γ(x)$由$γ(x):=\lim\limits_{n\to \infty}\frac{1}{n}\sum_{j=1}^n \log a_j(x)$定义。本文研究了Khintchine谱$\dim E_ξ$，其中$ E_ξ:=\{x\in [0,1):γ(x)=ξ\} (ξ\geq 0)$和$\dim$表示Hausdorff维数.特别是，我们证明了显着的事实，欣钦频谱$\dim E_ξ$，作为$ξ\in [0, +\infty)$的功能，既不是凹的，也不是凸的。从多重分形分析的通常观点来看，这是一个新现象。研究了由$γ^ϕ(x):=\lim\limits_{n\to\infty}\frac{1}{ϕ(n)} \sum_{j=1}^n \log a_j(x)$定义的快速Khintchine指数，其中$ϕ(n)$比$n$更快地趋于无穷大.在$ϕ$的某些正则条件下，证明了快Khintchine谱$\dim (\{x\in [0,1]: γ^ϕ(x)= ξ\}) $是一个常数函数.我们的方法也适用于其他光谱，如李雅普诺夫谱和快速李雅普诺夫谱。
英文摘要：
Assume that $x\in [0,1) $ admits its continued fraction expansion $x=[a_1(x), a_2(x),...]$. The Khintchine exponent $γ(x)$ of $x$ is defined by $γ(x):=\lim\limits_{n\to \infty}\frac{1}{n}\sum_{j=1}^n \log a_j(x)$ when the limit exists. Khintchine spectrum $\dim E_ξ$ is fully studied, where $ E_ξ:=\{x\in [0,1):γ(x)=ξ\} (ξ\geq 0)$ and $\dim$ denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum $\dim E_ξ$, as function of $ξ\in [0, +\infty)$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $γ^ϕ(x):=\lim\limits_{n\to\infty}\frac{1}{ϕ(n)} \sum_{j=1}^n \log a_j(x)$ are also studied, where $ϕ(n)$ tends to the infinity faster than $n$ does. Under some regular conditions on $ϕ$, it is proved that the fast Khintchine spectrum $\dim (\{x\in [0,1]: γ^ϕ(x)= ξ\}) $ is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.
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