On Khintchine exponents and Lyapunov exponents of continued fractions

Abstract

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.