TABLE DES MATIÈRES 5.3 Nouvelle preuve du théorème d'Oxtoby-Ulam . . . . . . . . 6 Transfert des propriétés ergodiques génériques de Auto(X, µ) vers Homeo(X, µ) 6.1 Densité des classes de conjugaison des apériodiques .
The goals of this paper are to obtain theoretical models of what happens when a computer calculates the rotation set of a homeomorphism, and to find a good algorithm to perform simulations of this rotation set. To do that we introduce the notion of observable rotation set, which takes into account the fact that we can only detect phenomenon appearing on positive Lebesgue measure sets; we also define the asymptotic discretized rotation set which in addition takes into account the fact that the computer calculates with a finite number of digits It appears that both theoretical results and simulations suggest that the asymptotic discretized rotation set is a much better approximation of the rotation set than the observable rotation set, in other words we need to do coarse roundoff errors to obtain numerically the rotation set.
Abstract. We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are C 0 -generic and have bounded rotational deviations (even for pseudo-orbits). The results hold both in the area-preserving setting and in the general setting. When the rotation set is stable, we give explicit estimates on the type of rationals that may appear as vertices of rotation sets in terms of the stability constants.
Abstract. We prove the genericity of the shadowing and periodic shadowing properties for both conservative and dissipative homeomorphisms on a compact connected manifold. Our proof is valid for topological manifolds and still holds in the dissipative case. As a consequence of this result, we establish the genericity of the specification property, the average shadowing property and the asymptotic average shadowing property in the conservative case.
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