In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and the strong separation condition. The main result of this paper is the existence of dimension maximizing Gibbs measures (Käenmäki measures). To prove this phenomena, we show that the Ledrappier-Young formula holds for Gibbs measures and we introduce a transversality type condition for the strong-stable directions on the projective space.

We consider one-step cocycles of $2 \times 2$ matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist and are characterized by their support. Under an additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.

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.