On Khintchine exponents and Lyapunov exponents of continued fractions
Résumé
Assume that $x\in [0,1) $ admits its continued fraction expansion $x=[a_1(x), a_2(x),\cdots]$. The Khintchine exponent $\gamma(x)$ of $x$ is defined by $\gamma(x):=\lim\limits_{n\to \infty}\frac{1}{n}\sum_{j=1}^n \log a_j(x)$ when the limit exists. Khintchine spectrum $\dim E_\xi$ is fully studied, where $ E_{\xi}:=\{x\in [0,1):\gamma(x)=\xi\} \ (\xi \geq 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 $\gamma^{\varphi}(x):=\lim\limits_{n\to\infty}\frac{1}{\varphi(n)} \sum_{j=1}^n \log a_j(x)$ 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.
Domaines
Systèmes dynamiques [math.DS]
Origine : Fichiers produits par l'(les) auteur(s)
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