On Khintchine exponents and Lyapunov exponents of continued fractions

Assume that x ∈[0,1) admits its continued fraction expansion x =[ a 1 ( x ), a 2 ( x ),…]. The Khintchine exponent γ ( x ) of x is defined by γ (x):=lim n→ ∞(1/n) ∑ ⱼ₌₁ⁿ log aⱼ(x) when the limit exists. The Khintchine spectrum dim E ξ is studied in detail, where E ξ :={ x ∈[0,1): γ ( x )= ξ }( ξ ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim E ξ , as a function of ξ ∈ [0, +∞ ) , is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by γ φ(x):=lim n→ ∞(1/(φ (n)))∑ ⱼ₌₁ⁿ log aⱼ(x) are also studied, where φ( n ) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({ x ∈[0,1]: γ φ ( x )= ξ }) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.