Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

Abstract

Given a symplectic surface (Σ, ω) of genus g ≥ 4, we show that the free group with two generators embeds into every asymptotic cone of (Ham(Σ, ω), dH), where dH is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds. * This paper was the outcome of the authors' work in the computational symplectic topology graduate student team-based research program held Hofer's geometryLet (M, ω) be a symplectic manifold. Given a smooth function H :where H t (p) := H(t, p). Let ϕ t H : M → M be the flow of the ODEẋ(t) = X t (x(t)), making sufficient assumptions to ensure that the flow is globally defined on the time interval [0, 1] (for example, we could take M to be compact). Inside the group Symp(M, ω) = {φ ∈ Diff(M ) : φ * ω = ω} of symplectomorphisms we have the subgroup of Hamiltonian diffeomorphisms Ham(M, ω), which consists of the time-one maps ϕ 1 H : M → M of flows as above. The group Ham(M, ω) is equipped with a geometrically meaningful bi-invariant metric introduced by Hofer. The resulting metric group is an important object of study in symplectic geometry. For φ ∈ Ham(M, ω), we define the Hofer norm φ H = inf H 1 0

Viterbo conjecture for Zoll symmetric spaces

“…This map is Lipschitz with respect to the L 1,∞ -distance on H = H M , and the bottleneck distance on the space barcodes of barcodes. This observation was used in [76], in [3,40,78,92,99,105] and more recently in [18,31,60,66,93,95] to produce various quantitative results in symplectic topology. Set barcodes ′ for the quotient space of barcodes with respect to the isometric R-action by shifts.…”

Lagrangian configurations and Hamiltonian maps

“…Recall [7] that the latter is a group equipped with a bi-invariant metric which, roughly speaking, reflects the large-scale geometry of (Ham(S 2 ), d Hofer ). For closed surfaces of genus ≥ 2, the asymptotic cone of the group of Hamiltonian diffeomorphisms contains a free group with two generators [4,11]. The construction is based on a chaotic dynamical system called the eggbeater map (see [49] for symplectic aspects of this map).…”

Persistence Modules with Operators in Morse and Floer Theory

“…[8,9,12,14,15,17,20,34,62]). Quite recently, persistence modules found applications in symplectic topology, see [1,29,47,52,56,61], with preludes in [6,31,54,55].…”

“…To do so, define a Z p persistence module whereφ λ is given by the egg-beater flow on Σ, with mixing parameter λ. Construction and detailed analysis of egg-beater flow are carried out in [1,47]. What we will use is that there exists a family of Hamiltonian flowsφ λ on Σ, depending on an unbounded increasing real parameter λ, along with a family of classes of free loops α λ on Σ which satisfy: 1) ϕ p λ has exactly 2 2p p-tuples of fixed points with same indices and actions {z, ϕ λ (z), .…”