Subexponentially increasing sums of partial quotients in continued fraction expansions

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S\_n(x)=\sum\_{j=1}^n a\_j(x)$, where $x=[a\_1(x), a\_2(x), \cdots ]$ is the continued fraction expansion of an irrational $x\in (0,1)$. Precisely, for an increasing function $\varphi: \mathbb{N} \rightarrow \mathbb{N}$, one is interested in the Hausdorff dimension of the sets$$E\_\varphi = \left\{x\in (0,1): \lim\_{n\to\infty} \frac {S\_n(x)} {\varphi(n)} =1\right\}.$$Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\exp(n^\gamma), \ \gamma \in [1/2, 1)$. We show that when $\gamma \in [1/2, 1)$, $E\_\varphi$ has Hausdorff dimension $1/2$. Thus, surprisingly, the dimension has a jump from $1$ to $1/2$ at $\varphi(n)=\exp(n^{1/2})$. In a similar way, the distribution of the largest partial quotient is also studied.Comment: 12 pages. More details for the proof of Theorem 1.2. are adde

Upper and lower fast Khintchine spectra in continued fractions

For an irrational number $x\in [0,1)$, let $x=[a\_1(x), a\_2(x),\cdots]$ be its continued fraction expansion. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to\infty$. The (upper, lower) fast Khintchine spectrum for $\psi$ is defined as the Hausdorff dimension of the set of numbers $x\in (0,1)$ for which the (upper, lower) limit of $\frac{1}{\psi(n)}\sum\_{j=1}^n\log a\_j(x)$ is equal to $1$. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.Comment: 13 pages. Motivation and details of proofs are adde

Diophantine approximation by orbits of Markov maps

In 1995, Hill and Velani introduced the shrinking targets theory. Given a dynamical system $([0,1],T)$, they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well-approximated by orbits $\{T^n x\}_{n\geq 0}$, where $T$ is an expanding Markov map with a finite partition supported by $[0,1]$. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.Comment: 24 pages, 3 figures; To appear in ETDS, 201

Rational map ax+1/x on the projective line over Q2\mathbb{Q}_2

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.Comment: 18 page

Multifractal analysis of some multiple ergodic averages for the systems with non-constant Lyapunov exponents

We study certain multiple ergodic averages of an iterated functions system generated by two contractions on the unit interval. By using the dynamical coding ${0,1}^{\mathbb{N}}$ of the attractor, we compute the Hausdorff dimension of the set of points with a given frequency of the pattern 11 in positions $k, 2k$.Comment: 13 page