A note on the Lyapunov exponent in continued fraction expansions

Authors

JIANZHONG CHENG
LU-MING SHEN

DOI

10.3906/mat-0807-16

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.

First Page

145

Last Page

152

Recommended Citation

CHENG, JIANZHONG and SHEN, LU-MING (2010) "A note on the Lyapunov exponent in continued fraction expansions," Turkish Journal of Mathematics: Vol. 34: No. 2, Article 1. https://doi.org/10.3906/mat-0807-16
Available at: https://journals.tubitak.gov.tr/math/vol34/iss2/1