Propriétés dynamiques génériques des homéomorphismes conservatifs

Abstract

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 .

Spatial discretizations of generic dynamical systems

“…King, see [GK98]). Interested readers can consult the surveys [AP00] and [Gui12]. Let us give a sketch of the proof of genericity of transitivity (probably one of the simplest proofs): considering a conservative homeomorphism f , -we begin by breaking the dynamics f in discretizing the phase space and applying Lax's theorem (see [Lax71] and theorem 5.5 page 70), which asserts that a conservative homeomorphism is arbitrarily close to a cyclic permutation of a grid of the manifold having some good properties; -we then rebuild a conservative homeomorphism from this cyclic permutation, by a C 0 closing lemma (Proposition 3.3 page 44).…”

“…This theorem was later improved by S. Alpern in [Alp76,Alp78] to establish generic properties of conservative homeomorphisms (see also [DKP93,KM97] for a generalisation and some simulations in dimension 1). This theorem is now one of the keystones of the theory of generic properties of homeomorphisms (see [Gui12]), together with other theorems of approximation by permutations 3 .…”

“…In fact, discretizations on arbitrary manifolds can be easily obtained from discretization grids on the cube I n (for example the grids E 0 N ) by applying the Oxtoby-Ulam theorem (see [OU41] or [Gui12,Corollaire 1.4]). This theorem asserts that every smooth connected compact manifold of dimension n, possibly with boundary, equipped with a good measure λ, is obtained from the unit cube I n by gluing together some parts of the boundary of the cube.…”

“…3. However, some properties can be proved in different ways: for instance Corollary 5.20 can be proved by inserting horseshoes in the dynamics of a given homeomorphism, using the local modification theorem (Theorem 5.40, for a presentation of the technique in another context see [Gui12,Section 3.3]); also a variation of Corollary 5.15 can be shown in perturbing any given homeomorphism such that it has a periodic orbit whose distance to the grid E N is smaller than the modulus of continuity of f , so that the actual periodic orbit and the discrete orbit fit together.…”

“…As a compact metric set can be seen as a "finite set up to ε", Lax's theorem allows us to see a homeomorphism as a "cyclic permutation up to ε". Genericity of transitivity in Homeo(X, λ) follows easily from Lax's theorem together with the finite map extension proposition (see [AP00] or part 2.4 of [Gui12]). In our case, applying Theorem 5.2, we deduce that infinitely many discretizations of a generic homeomorphism are cyclic permutations.…”