2016/03/21 by Pierre‐Antoine Guihéneuf, Guihéneuf, Pierre-Antoine
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1603.06480
openalex publication_date 2016/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 fN (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 fN 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 C1-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations fN, 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 fN : its dynamics strongly depends on the order N. To detect some dynamical features of f, we have to consider all the discretizations fN when N goes through \mathbf N. The second part deals with the linear case, which plays an important role in the study of C1-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.