Journals / Turkish Journal of Mathematics / 2010 / Cilt: 34 - Sayı: 2
A note on the Lyapunov exponent in continued fraction expansions
- Pages
- 145–151
- DOI
- —
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.
Özet
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.