Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$ , there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$ . We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$ , where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$ -transformation.

We analyse dynamical properties of the negative beta-transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta-transformation, the density of the absolutely continuous invariant measure of the negative beta-transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β>1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (−β)-expansion of 1 when β tends to 1 is related to the Thue–Morse sequence. A consequence of the exactness is that every Yrrap number, which is a β>1 such that the (−β) -expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

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 {Tⁿx}_n≥0, where Tis 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.

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 = [a₁(x), a₂(x), . . .] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : $\mathbb{N}$ → $\mathbb{N}$ , one is interested in the Hausdorff dimension of the set 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^γ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), E_ϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ(n) = exp(n^1/2). In a similar way, the distribution of the largest partial quotient is also studied.