I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions.[-]

​An almost Anosov flow is a flow having continuous flow-invariant splitting of the tangent bundle with exponential expansion/contraction in the unstable/stable direction, except for a finite number (in our case a single) periodic orbits. Roughly, almost Anosov flows are perturbed Anosov flows, where the perturbation is local around these periodic orbits, making them neutral. For this type of flows, we obtain limit theorems (stable, standard and non-standard CLT) for a large class of (unbounded) observables. I will present these results stressing on the method of proof. This is joint work with H. Bruin and M. Todd.[-]