A note on the Lyapunov exponent in continued fraction expansions

Authors : Jianzhong CHENG

Pages : 145-152

View : 14 | Download : 8

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let T:[0,1); \to [0,1); be the Gauss transformation. For any irrational x \in [0,1);, the Lyapunov exponent ainsert ignore into journalissuearticles values(x); of x is defined as ainsert ignore into journalissuearticles values(x);=\limn\to\infty\frac{1}{n} \log |insert ignore into journalissuearticles values(Tn);`insert ignore into journalissuearticles values(x);|. By Birkoff Average Theorem, one knows that ainsert ignore into journalissuearticles values(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 |insert ignore into journalissuearticles values(Tn);`insert ignore into journalissuearticles values(x);| does not exist\} carries full Hausdorff dimension.
Keywords : Continued fractions, Lévy constant, Hausdorff dimension