Turkish Journal of Mathematics
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.
DOI
10.3906/mat-0807-16
Keywords
Continued fractions, Lévy constant, Hausdorff dimension.
First Page
145
Last Page
152
Recommended Citation
CHENG, J, & SHEN, L (2010). A note on the Lyapunov exponent in continued fraction expansions. Turkish Journal of Mathematics 34 (2): 145-152. https://doi.org/10.3906/mat-0807-16
