A note on the Lyapunov exponent in continued fraction expansions

作者：Cheng, Jianzhong^*; Shen, Lu-Ming

来源：Turkish Journal of Mathematics, 2010, 34(2): 145-151.

DOI：10.3906/mat-0807-16

• Continued fractions
• Levy constant
• Hausdorff dimension

Let T : [0, 1) --> [0, 1) be the Gauss transformation. For any irrational x is an element of [0, 1), the Lyapunov exponent alpha(x) of x is defined as
alpha(x) = lim(n-->infinity) 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 is an element of [0, 1) : lim(n-->infinity) 1/n log |(T(n))'(x)| does not exist}
carries full Hausdorff dimension.

• 出版日期2010
• 单位湖南农业大学