A note on the Lyapunov exponent in continued fraction expansions

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.

Anahtar Kelimeler:

Continued fractions, Lévy constant, Hausdorff dimension.

A note on the Lyapunov exponent in continued fraction expansions

Keywords:

Continued fractions, Lévy constant, Hausdorff dimension.,

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