Dynamical properties of spatial discretizations of a generic homeomorphism

Abstract

This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412-414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

Physical measures of discretizations of generic diffeomorphisms

“…Actually, when we perform a lot of simulations, we realize that there are also big variations of the behaviour of the measures (Figure 14): the measure is often well distributed in the torus, and sometimes quite singular with respect to Lebesgue measure (as can be seen in Figure 11). This behaviour is almost identical to that observed in the C 0 case in the neighbourhood of A (see [Gui15c]). neighbourhoods of the boundaries of these charts 'counts for nothing' from the Lebesgue measure viewpoint.…”

“…We can test whether this property can be observed in simulations or not. T 2 , where g 1 is a small C 1 perturbation of the identity, is way smoother than in the C 0 case (compare Figure 12 with [Gui15c]). Indeed, the measure µ…”

“…However, one can hope that their behaviour is not as erratic as for generic conservative homeomorphisms, where they accumulate on the whole set of f -invariant measures (see [Gui15c,§4.6]). Indeed, [Gui15a] shows that a simple dynamical invariant associated to each discretization f N (the degree of recurrence) converges to 0 as N tends to +∞ for a generic conservative C 1 -diffeomorphism, while it accumulates on the whole segment [0, 1] for generic conservative homeomorphisms; this shows that the discretizations of generic conservative C 1 -diffeomorphisms and homeomorphisms might have quite different dynamical characteristics.…”

Extensions with shrinking fibers

“…Often, physical measure are said to be the ones that can be seen in practice, given they drive the behavior of a positive proportion of the points. However, note that in some cases Guihéneuf [Gui15] has shown that non-physical measures could be actually observed.…”

Generic properties of extensions

“…Since then, much research has been done in finding "typical" properties of measure-preserving dynamical systems. See, e.g., [28], [26], [1], [27], [2], [5], [3], [12], [4], [33], [21].…”