On Khintchine exponents and Lyapunov exponents of continued fractions

On Khintchine exponents and Lyapunov exponents of continued fractions

Assume that x ∈ [0, 1) admits its continued fraction expansion x = [a1(x), a2(x), · · · ]. The Khintchine exponent γ(x) of x is defined by γ(x) := lim n→∞ 1 n Pn j=1 log aj(x) when the limit exists. Khintchine spectrum dimEξ is fully studied, where Eξ := {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spect...

Lectures on Lyapunov exponents

Lyapunov exponents, dual Lyapunov exponents, and multifractal analysis.

It is shown that the multifractal property is shared by both Lyapunov exponents and dual Lyapunov exponents related to scaling functions of one-dimensional expanding folding maps. This reveals in a quantitative way the complexity of the dynamics determined by such maps. (c) 1999 American Institute of Physics.

Lyapunov Exponents

The analysis of potentially chaotic behavior in biological and biomedical phenomena has attracted great interest in recent years (1–6). Although no universally accepted mathematical definition of the term chaos exists, Strogatz (7) provides a working definition as ‘‘aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.’’ Aperiodic long-...

Lyapunov Exponents

We are interested in iterates of the logistic map T : [0, 1] → [0, 1] defined by