- Turkish Journal of Mathematics
- Vol: 34 Issue: 2
- A note on the Lyapunov exponent in continued fraction expansions
A note on the Lyapunov exponent in continued fraction expansions
Authors : Jianzhong Cheng
Pages : 145-152
View : 7 | Download : 6
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :Let T:[0,1) \to [0,1) be the Gauss transformation. For any irrational x \in [0,1), the Lyapunov exponent a(x) of x is defined as a(x)=\limn\to\infty\frac{1}{n} \log |(Tn)'(x)|. By Birkoff Average Theorem, one knows that a(x) exists almost surely. However, in this paper, we will see that the non-typical set \{x\in [0,1):\limn\to\infty\frac{1}{n} \log |(Tn)'(x)| does not exist\} carries full Hausdorff dimension.Keywords : Continued fractions, Lévy constant, Hausdorff dimension.