On Khintchine exponents and Lyapunov exponents of continued fractions
On Khintchine exponents and Lyapunov exponents of continued fractions
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DOI：
10.1017/s0143385708000138
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发表时间：
2008-02
影响因子：
0.9
通讯作者：
A. Fan;Lingmin Liao;Bao-Wei Wang;Jun Wu
中科院分区：
文献类型：
--
作者：
A. Fan;Lingmin Liao;Bao-Wei Wang;Jun Wu
Abstract Assume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by $\gamma (x):=\lim _{n\to \infty }({1}/{n}) \sum _{j=1}^n \log a_j(x)$ when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of $\xi \in [0, +\infty )$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma ^{\varphi }(x):=\lim _{n\to \infty }({1}/({\varphi (n)}))\sum _{j=1}^n \log a_j(x)$ are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.
Abstract Assume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by $\gamma (x):=\lim _{n\to \infty }({1}/{n}) \sum _{j=1}^n \log a_j(x)$ when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of $\xi \in [0, +\infty )$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma ^{\varphi }(x):=\lim _{n\to \infty }({1}/({\varphi (n)}))\sum _{j=1}^n \log a_j(x)$ are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.