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 $\gamma (x)$ of $x$ is defined by $\gamma (x):=\underset{n\to \infty }{\lim }\frac{1}{n}{\sum }_{j=1}^n\log a_j(x)$ when the limit exists. Khintchine spectrum $\dim E_{\xi }$ is fully studied, where $E_{\xi }:=\{x\in [0,1):\gamma (x)=\xi \}(\xi \ge 0)$ and $\dim$ denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum $\dim E_{\xi }$, as 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 ${\gamma }^{\varphi }(x):=\underset{n\to \infty }{\lim }\frac{1}{\varphi (n)}{\sum }_{j=1}^n\log a_j(x)$ are also studied, where $\varphi (n)$ tends to the infinity faster than $n$ does. Under some regular conditions on $\varphi$, it is proved that the fast Khintchine spectrum $\dim (\{x\in [0,1]:{\gamma }^{\varphi }(x)=\xi \})$ is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.

37 pages, 5 figures, accepted by Ergodic Theory and Dyanmical Systems