A note on the Lyapunov exponent in continued fraction expansions

Authors: JIANZHONG CHENG, LU-MING SHEN

Abstract: Let T:[0,1) \to [0,1) be the Gauss transformation. For any irrational x \in [0,1), the Lyapunov exponent \alpha(x) of x is defined as \alpha(x)=\lim_{n\to\infty}\frac{1}{n} \log |(T^n)'(x)|. By Birkoff Average Theorem, one knows that \alpha(x) exists almost surely. However, in this paper, we will see that the non-typical set \{x\in [0,1):\lim_{n\to\infty}\frac{1}{n} \log |(T^n)'(x)| does not exist\} carries full Hausdorff dimension.

Keywords: Continued fractions, Lévy constant, Hausdorff dimension.

Full Text: PDF