Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants

Abstract

We use the Hofer norm to show that all Hamiltonian diffeomorphisms with compact support in R 2n that displace an open connected set with a nonzero Hofer-Zehnder capacity move a point farther than a capacitydependent constant. In R 2 , this result is extended to all compactly supported area-preserving homeomorphisms. Next, using the spectral norm, we show the result holds for Hamiltonian diffeomorphisms on closed surfaces. We then show that all area-preserving homeomorphisms of S 2 and RP 2 that displace the closure of an open connected set of fixed area move a point farther than an area-dependent constant.2010 Mathematics Subject Classification. Primary: 57R17. Key words and phrases. Spectral Norm, Displaced Disks, Hofer Norm. We would like to thank Dmitri Burago for informing us of the problem of displacing sets on the sphere and to again thank Dmitri Burago, Sergei Ivanov, and Leonid Poltervich for the idea to use the spectral norm to solve this problem. We would also like to thank Misha Guysinsky for his guidance and for many useful discussions and comments. We thank Sobhan Seyfaddini for suggesting that his results in [10] would allow us to prove the result on the cotangent bundle.

The action spectrum and C^0 symplectic topology

“…In the specific case where a = [M ], b = [pt], the function γ([M ], [pt]; •) : Ham(M, ω) → R induces a non-degenerate norm on Ham(M, ω) which is referred to as the spectral norm and is simply denoted by γ(•). Over the past decade, with the expansion of C 0 symplectic topology, the question of C 0 continuity of this norm on closed symplectic manifolds, and whether it extends to Hamiltonian homeomorphisms, has received much attention (see [39,40,48,49,50,8]) and has only been answered in the case of surfaces in [49]. Our main result settles this question for any closed, connected and symplectically aspherical manifold.…”

“…An important feature of almost conjugacy is that in Rokhlin groups any two elements are almost conjugate, and hence the relation is trivial for such groups. 8 The two theorems below were first proven in the two-dimensional setting in [32]. Here, we extend them to higher dimensional symplectically aspherical manifolds.…”

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

“…Likewise, on the torus, since Ham(T 2 ) is simply connected ( [34], Section 7.2), it depends only on φ 1 if we restrict ourselves to Hamiltonian isotopies. 5 The basic result on unlinked sets is the following. Theorem 8.…”

“…2 Spectral invariants have had many important and interesting applications in symplectic topology and dynamical systems; see for example [7,8,13]. A recently discovered application which has largely motivated this article is a simple solution to the displaced disks problem of Béguin, Crovisier and Le Roux: using the spectral invariant c one can show that arbitrarily C 0 -small area preserving homeomorphisms of a closed surface can not displace disks of a given area; see [43,5].…”