We shall investigate links between the differential dynamic of closed manifolds and the symbolic dynamic of their fundamental group. Despite appearances, this is a talk about the arithmetic groups, and all the required notions of differential geometry will be carefully explained.

Let $A$ be a finite alphabet, and let $A^{\omega}$ be the set of semi-infinite words over $A$. An action of a group $G$ on $A^{\omega}$ is called $\it{similar}$ if for any $g\in G$, $a\in A$ there is $b\in A$ and $g_{a}\in G$ such that

$g\cdot (a\ast w) = b\ast g_{a}\cdot \omega$, for all $\omega\in A^{\omega}$,

where $\cdot$ is the symbol of the action and $\ast$ is the concatenation. The action is called $\it{minimal}$ if all orbits are dense (note that $A^{\omega} = \lim_{\leftarrow} A^{n}$ has a natural topology).

Recall that a diffeomorphism $\theta$ of a closed manifold $M$ is called Anosov if there is a $d\theta-invariant$ decomposition $TE = E_{u}\oplus E_{s}$ such that $d\theta|_{E_{s}}$ and $d\theta^{-1}|_{E_{u}}$ are contracting, with respect to some riemannian metric.

One of our two conjectures involves a closed manifold $M$ endowed with an Anosov diffeomorphism.

$Conjecture.$ There exists a minimal, free and self-similar action of $\pi1(M)$ on $A^{\omega}$ for some finite alphabet $A$.

$Theorem.$ Let $\overset{\sim}{M}$ be the universal cover of $M$. Assume that

(i) $\pi(M)$ is torsion free and virtually nilpotent,

(ii) $H\ast(\overset{\sim}{M}, \mathbb{R})$ is finite dimensional.

Then the conjecture holds for $M$.

This implies that our conjecture is a corollary of the classical Anosov-Smale conjecture. A second set of conjecture/result involves manifolds with an affine structure, in the spirit of Milnor's conjecture.