A note on the Lyapunov exponent in continued fraction expansions

Let T : [0, 1) → [0, 1) be the Gauss transformation. For any irrational x ∈ [0, 1), the Lyapunov exponent α(x) of x is defined as alpha(x)=lim_{ntoinfty}frac{1}{n} log |(T^n) (x)|. By Birkoff Average Theorem, one knows that α(x) exists almost surely. However, in this paper, we will see that the non-typical set{xin [0,1):lim_{ntoinfty}frac{1}{n} log |(T^n) (x)| does not exist} carries full Hausdorff dimension