On Khintchine exponents and Lyapunov exponents of continued fractions

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