We will first explain three properties of a Zariski dense subsemigroups of a semisimple real Lie group:

- the existence of loxodromic elements,

- the convexity and non degeneracy of the limit cone,

- the density of the group spanned by the Jordan projection.

We will then explain how to deduce from them three limit theorems for a random walk on this semigroup:

- the law of large numbers,

- the central limit theorem,

- the local limit theorem.

This minicourse is an introduction to the group of conformal transformations of the sphere. We will focus on its action on the space of non-singleton compact subsets of the sphere.
1. Conformal autosimilarity: we will see that the compact subsets  whose orbit is closed are exactly the limit sets of convex cocompact subgroups.
2. Quasicircles: we  will see that the compact subsets whose orbit closure contains only Jordan curves are exactly the quasicircles.
3. Minimal orbits: we will construct compact subsets of the sphere whose orbit is not closed but whose orbit closure is minimal.

This minicourse is an introduction to the group of conformal transformations of the sphere. We will focus on its action on the space of non-singleton compact subsets of the sphere.
1. Conformal autosimilarity: we will see that the compact subsets  whose orbit is closed are exactly the limit sets of convex cocompact subgroups.
2. Quasicircles: we  will see that the compact subsets whose orbit closure contains only Jordan curves are exactly the quasicircles.
3. Minimal orbits: we will construct compact subsets of the sphere whose orbit is not closed but whose orbit closure is minimal.