On Khintchine exponents and Lyapunov exponents of continued fractions

Abstract

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 $\gamma (x):=\lim _{n\to \infty }({1}/{n}) \sum _{j=1}^n \log a_j(x)$ when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, 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 spectrum dim Eξ, as a function of $\xi \in [0, +\infty )$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma ^{\varphi }(x):=\lim _{n\to \infty }({1}/({\varphi (n)}))\sum _{j=1}^n \log a_j(x)$ 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]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

Upper and lower fast Khintchine spectra in continued fractions

“…In contrast to continued fractions, in symbolic dynamical systems of finitely many symbols there is no fast spectrum since the Birkhoff averages are usually bounded (for example, when φ is continuous). Thus the fast spectra studied in this note and in [4,14,15,10] are new subjects for continued fractions and can have generalization in symbolic dynamical systems of infinitely many symbols.…”

“…The Hausdorff dimension (denoted by dim H ) of the level sets E(α) := x ∈ [0, 1) : lim n→∞ log a 1 (x) + · · · + log a n (x) n = α , α > 0, was calculated in [4]. It turns out that the Khintchine spectrum, i.e., the function α → dim H E(α) is a real-analytic curve increasing on [0, ξ 0 ] and decreasing on (ξ 0 , ∞).…”

Divergence points with fast growth orders of the partial quotients in continued fractions

“…The result in the following lemma can be found in our former work [9]. But here a slightly stronger version is needed, moreover, a concise technique is illuminated.…”

“…It should be mentioned that the multifractal analysis concerning the partial quotients are given, (I) on the convergence points, by A. H. Fan and L. M. Liao and the authors [9], where a rather integral determination and characterization on the spectrum function…”

Large deviations for the local fluctuations of random walks

“…; a n ]}. It is clear that the question reduces to studying n −1 log ∑ (a 1 ,...,a n )∈(N + ) n (a 1 · · · a n ) λ |I a 1 ,...,a n |; this sequence converges for λ < 1 to a limit (λ) analytic in λ (see Section 4 of [17]). Consequently, Theorem 2.3, Corollary 2.1 and Theorem 2.4 provide large deviation properties for the local fluctuations of log(a 1 (t)) + · · · + log(a n (t)) almost everywhere with respect to the Lebesgue measure.…”