Spatial discretizations of generic dynamical systems
  How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations? To try to answer this question, we study in this manuscript a model reflecting what happens when the orbits of a discrete time system $f$ (for example an homeomorphism) are computed numerically . The computer working in finite numerical precision, it will replace $f$ by a spacial discretization of $f$, denoted by $f_N$ (where the order $N$ of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps $f_N$ for a generic system $f$ and $N$ going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or $C^1$-diffeomorphisms).   The first part of this manuscript is devoted to the study of the dynamics of the discretizations $f_N$, when $f$ is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of $f$ from that of a single discretization $f_N$ : its dynamics strongly depends on the order $N$. To detect some dynamical features of $f$, we have to consider all the discretizations $f_N$ when $N$ goes through $\mathbf N$.   The second part deals with the linear case, which plays an important role in the study of $C^1$-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part, though weaker and harder to prove. 
Pierre-Antoine Guih\'eneuf
math.DS