Partial hyperbolicity or dense elliptic periodic points for 𝐶¹-generic symplectic diffeomorphisms

Abstract

Abstract. We prove that if a symplectic diffeomorphism is not partially hyperbolic, then with an arbitrarily small C 1 perturbation we can create a totally elliptic periodic point inside any given open set. As a consequence, a C 1 -generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points. This extends the similar results of S. Newhouse in dimension 2 and M.-C. Arnaud in dimension 4. Another interesting consequence is that stably ergodic symplectic diffeomorphisms must be partially hyperbolic, a converse to Shub-Pugh's stable ergodicity conjecture for the symplectic case.

Billiards: A singular perturbation limit of smooth Hamiltonian flows

“…From a mathematical point of view, the appearance of islands of stability is natural in Hamiltonian systems which are not hyperbolic or partially hyperbolic, see Ref. 45 and references therein for the (C 1 )-version of this conjecture. However, a specific family of systems like (1), limiting to the hyperbolic Sinai billiards, may turn out to be non-generic (see the introduction in Ref.…”

Shades of hyperbolicity for Hamiltonians

“…Long after Newhouse's proof, Arnaud (see [2]) proved the 4-dimensional version of this result, namely that C 1 -generic symplectomophisms are Anosov, partially hyperbolic or have dense elliptic periodic points. Finally, Saghin and Xia ( [51]) proved the same result but for any dimension completing the program for the discrete case. In this paper, and among other issues, we developed, in Theorem 4, the approach for the continuous-time case of Saghin-Xia's theorem.…”

Homoclinic intersections of symplectic partially hyperbolic systems with 2D center

“…Moreover, it is proved in [SX06] that symplectic partially hyperbolic maps are symmetric. That is, one can take ν = ν and γ = γ in Definition 2.1.…”