The autonomous norm on $${\text {Ham}}\left( {\mathbf R}^{2n} \right) $$ Ham
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The autonomous norm on Ham R2n is bounded Michael Brandenbursky1 · Jarek Ke˛ dra2,3
Received: 14 March 2016 / Accepted: 6 May 2016 / Published online: 16 July 2016 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract We prove that the autonomous norm on the group of compactly supported Hamiltonian diffeomorphisms of the standard R2n is bounded. Keywords Hamiltonian diffeomorphisms · Autonomous norm Résumé Nous montrons que la norme autonome sur le groupe des difféomorphismes Hamiltoniens à support compact de R2n standard est bornée. Mathematics Subject Classification Primary 53; Secondary 57 Let (M, ω) be a symplectic manifold and let Ham(M, ω) be the group of compactly supported Hamiltonian diffeomorphisms of (M, ω). Recall that a Hamiltonian diffeomorphism f is the time-one map of the flow generated by the vector field X Ft defined by ω(X Ft , −) = d Ft . Here F : M × S 1 → R is a smooth compactly supported function and F(x, t) = Ft (x) (see [9, Section 5.1] for details). The function F is called a Hamiltonian of f . If F does not depend on time then f is called autonomous. It is known that every Hamiltonian diffeomorphism is a product of autonomous ones [3]. The autonomous norm on Ham(M, ω) is defined by: f = min{k ∈ N | f = a1 . . . ak , whereai is autonomous}. It is a conjugation invariant norm and is known to be unbounded on the group of compactly supported Hamiltonian diffeomorphisms of an oriented surface of finite area [2–4,6].
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Michael Brandenbursky [email protected] Jarek Ke˛dra [email protected]
1
Ben Gurion University, Beer Sheva, Israel
2
University of Aberdeen, Aberdeen, UK
3
University of Szczecin, Szczecin, Poland
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M. Brandenbursky, J.Ke˛dra
This paper is concerned with the group Ham(R2n ) of compactly supported Hamiltonian diffeomorphisms of the Euclidean space equipped with the standard symplectic form. We prove the following result. Theorem The diameter of the autonomous norm on Ham R2n is bounded above by 3. Proof Let f ∈ Ham R2n . Let f = am . . . a1 , where ai ∈ Ham R2n are autonomous diffeomorphisms with compactly supported Hamiltonian functions Fi : R2n → R. Let B(r ) be the Euclidean ball centered at the origin, of radius r > 0 large enough so that it contains the union of the supports of the functions Fi . Lemma There exists an autonomous diffeomorphism h ∈ Ham R2n such that h i (B(r )) ∩ h j (B(r )) = ∅ for 0 ≤ i = j ≤ m. The statement of the lemma means that h displaces the ball B(r ) m times. It follows from [5, Lemma 2.6] that there exists g ∈ Ham R2n such that the following equality holds: 2
m
h , f = am . . . a1 = [h, g]a1h a2h . . . am
where aih = h i ai h −i and h is the diffeomorphism from the Lemma. Observe that, since the supports of Fi ◦ h −i are pairwise disjoint for i ∈ {1, . . . , m}, we obtain that the composition 2 h m is autonomous with the Hamiltonian function equal to a1h a2h . . . am i
F1 ◦ h −1 + F2 ◦ h −2 + · · · + Fm ◦ h −m . Since the commutator [h, g] = h · (h −1 )g is a pro
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