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Volume Growth in the Component of the Dehn–Seidel Twist

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Volume Growth in the Component of the Dehn–Seidel Twist VOLUME GROWTH IN THE COMPONENT OF THE DEHN–SEIDEL TWIST URS FRAUENFELDER AND FELIX SCHLENK Abstract. We consider an entropy-type invariant which mea- sures the polynomial volume growth of submanifolds under the iterates of a map, and we show that this invariant is at least 1 for every diffeomorphism in the symplectic isotopy class of the Dehn– Seidel twist. Contents 1. Introduction and main results 1 2. Proofs 5 2.1. Idea of the proof of Theorem 1 5 2.2. P -manifolds 6 2.3. Twists 7 2.4. LagrangianFloerhomology 10 m 2.5. Computation of HF∗ (ϑ (Dx),Dy) 15 2.6. EndoftheproofofTheorem1 23 2.7. Proof of Corollary 1 25 2.8. A remark on smoothness 26 2.9. Differential topology of twists 26 References 29 1. Introduction and main results The complexity of a compactly supported smooth diffeomorphism ϕ of a smooth manifold M can be measured in the following way: Fix a Riemannian metric g on M. For i ∈ {1,..., dim M} denote by Σi the set of smooth embeddings σ of the cube Qi = [0, 1]i into M, and i by µg(σ) the volume of σ (Q ) ⊂ M computed with respect to the measure on σ (Qi) induced by g. Following [21], we define for each i ∈ Date: January 12, 2005. 2000 Mathematics Subject Classification. Primary 53D35, Secondary 37B40, 53D40. 1 2 Volume growth in the component of the Dehn–Seidel twist {1,..., dim M} the i-dimensional slow volume growth si(ϕ) ∈ [0, ∞] by n log µg (ϕ (σ)) si(ϕ) = sup lim inf . n→∞ σ∈Σi log n This is the polynomial volume growth of the iterates of the most dis- torted smooth i-dimensional family of initial data. Since ϕ is compactly supported, si(ϕ) does not depend on the choice of g, and sdim M (ϕ)=0. Uniform lower bounds of s1(ϕ) were first obtained in a beautiful paper of Polterovich [29] for a class of symplectomorphisms ϕ =6 id in the identity component Symp0(M) of the group of symplectomorphisms of a closed symplectic manifold with vanishing second homotopy group. E.g., s1(ϕ) ≥ 1 for every symplectomorphism ϕ ∈ Symp0 (M,ω) \{id} of a closed oriented surface of genus ≥ 2, and 1 if d =1, s1(ϕ) ≥ 1 2 if d ≥ 2, for every non-identical Hamiltonian diffeomorphism of the standard 2d- dimensional torus. We endow the cotangent bundle T ∗B over a closed base B with the canonical symplectic structure ω = dλ, and denote c ∗ c ∗ by Symp0 (T B) the identity component of the group Symp (T B) of compactly supported C∞-smooth symplectomorphisms of (T ∗B,dλ). Combining a result in [15] with the arguments in [29], one finds c ∗ Remark 1 For every non-identical symplectomorphism ϕ ∈ Symp0 (T B) it holds true that s1(ϕ) ≥ 1. We refer to [16] for a proof. In this paper, we address a question of Polterovich in [30] and study the slow volume growth of certain compactly supported symplectomor- phisms outside the identity component. The spaces we shall consider are the cotangent bundles (T ∗B,dλ) over compact rank one symmetric spaces (CROSSes, for short), and the diffeomorphisms are Dehn twist like symplectomorphisms. These maps were introduced to symplectic topology by Arnol’d [2] and Seidel [35, 36]. They play a prominent role in the study of the symplectic mapping class group of various symplectic manifolds, [22, 34, 35, 36, 41], and generalized Dehn twists along spheres can be used to detect symplectically knotted Lagrangian spheres, [35, 36], and (partly through their appearance in Seidel’s long exact sequence in symplectic Floer homology, [40]) are an important ingredient in attempts to prove Kontsevich’s homological mirror sym- metry conjecture, [22, 37, 38, 39, 42, 43]. 3 Let (B,g) be a CROSS of dimension d, i.e., B is a sphere Sd, a d n n C À projective space ÊP , P , P , or the exceptional symmetric space 2 F4/ Spin9 diffeomorphic to the Cayley plane CaP . All geodesics on (B,g) are embedded circles of equal length. We define ϑ to be the compactly supported diffeomorphism of T ∗B whose restriction to the cotangent bundle T ∗γ ⊂ T ∗B over any geodesic circle γ ⊂ B is the square of the ordinary left-handed Dehn twist along γ depicted in Fig- ure 1. ϑ|T ∗γ γ T ∗γ Figure 1. The map ϑ|T ∗γ . We call ϑ a twist. A more analytic description of twists is given in Section 2.3. It is known that twists are symplectic, and that the class of a twist generates an infinite cyclic subgroup of the mapping class c ∗ group π0 (Symp (T B)), see [36]. A d-dimensional submanifold L of T ∗B is called Lagrangian if ω vanishes on T L × T L. Lagrangian submanifolds play a fundamental role in symplectic geometry. For each ϕ ∈ Sympc (T ∗B) we therefore also consider its Lagrangian slow volume growth log µ (ϕn(σ)) l(ϕ) = sup lim inf g σ∈Λ n→∞ log n where Λ is the set smooth embeddings σ : Qd ֒→ T ∗B for which σ Qd ∗ is a Lagrangian submanifold of T B. Of course, l(ϕ) ≤ sd(ϕ). As we m m shall see in Section 2.3, si(ϑ )= l(ϑ ) = 1 for every i ∈{1,..., 2d − 1} and every m ∈ Z \{0}. Theorem 1 Let B be a d-dimensional compact rank one symmetric space, and let ϑ be the twist of T ∗B described above. Assume that c ∗ m c ∗ ϕ ∈ Symp (T B) is such that [ϕ] = [ϑ ] ∈ π0 (Symp (T B)) for some m ∈ Z \{0}. Then sd(ϕ) ≥ l(ϕ) ≥ 1. 4 Volume growth in the component of the Dehn–Seidel twist 2n n Theorem 1 is of particular interest if B is S or CP , n ≥ 1, since in these cases it is known that (a power of) ϑ can be deformed to the identity through compactly supported diffeomorphisms, see [23, 36] and 2n+1 2n+1 Proposition 2.23 below. If B is S or ÊP , then the variation homomorphism of ϑ does not vanish, and so Theorem 1 already holds for homological reasons in this case, see Corollary 2.21 below. Twists can be defined on the cotangent bundle of any Riemannian manifold with periodic geodesic flow, and we shall prove Theorem 1 for all known such manifolds. In the case that B is a sphere Sd, one can use the fact that all geodesics emanating from a point meet again in the antipode to see that the twist ϑ admits a square root τ ∈ Sympc T ∗Sd . For d = 1, τ is the ordinary left-handed Dehn twist along a circle, and for d ≥ 2 it is (the inverse of) the generalized Dehn twist thoroughly studied in [34, 35, 36, 41]. Given any great circle γ in Sd, the restriction of τ to T ∗γ ⊂ T ∗Sd is the ordinary left-handed Dehn twist along γ depicted in Figure 2. τ|T ∗γ γ T ∗γ Figure 2. The map τ|T ∗γ . Corollary 1 Let τ be the (generalized) Dehn twist of T ∗Sd described above, and assume that ϕ ∈ Sympc T ∗Sd is such that [ϕ] = [τ m] ∈ c ∗ d π0 Symp T S for some m ∈ Z \{ 0}. Then sd(ϕ) ≥ l(ϕ) ≥ 1. 2 c ∗ d Since [τ ] = [ϑ] has infinite order in π0 Symp T S , so has [τ]. In c ∗ 2 the case d = 2 it is known that [τ] generates π0 (Symp (T S )), see [34, 41]. Remark 1 and Corollary 1 thus give a nontrivial uniform lower bound of the slow volume growth s(ϕ) = max si(ϕ) i for each ϕ ∈ Sympc(T ∗S2) \{id}. 5 Following Shub [44], we consider a symplectomorphism ϕ ∈ Sympc (T ∗B) as a best diffeomorphism in its symplectic isotopy class if ϕ minimizes m m both s(ϕ) and l(ϕ). We shall show that si(τ ) = l(τ ) = 1 and m m si(ϑ )= l(ϑ ) = 1 for every i ∈{1,..., 2d − 1} and every m ∈ Z \{0}. In view of Theorem 1 and Corollary 1, the twists τ m and ϑm are then best diffeomorphisms in their symplectic isotopy classes. Acknowledgements. This paper owes very much to Leonid Polterovich and Paul Seidel. We cordially thank both of them for their help. We are grateful to Yuri Chekanov, Anatole Katok, Nikolai Krylov, Janko Latschev and Dietmar Salamon for useful discussions. Much of this work was done during the stay of both authors at the Symplectic Topol- ogy Program at Tel Aviv University in spring 2002 and during the second authors stay at FIM of ETH Z¨urich and at Leipzig University. We wish to thank these institutions for their kind hospitality, and the Swiss National Foundation and JSPS for their generous support. 2. Proofs We shall first outline the proof of Theorem 1. We then describe the known Riemannian manifolds with periodic geodesic flow and define twists on such manifolds. We next prove Theorem 1 and Corollary 1, and notice that these results continue to hold for C1-smooth symplec- tomorphisms. We finally study twists from a topological point of view. 2.1. Idea of the proof of Theorem 1. Consider the cotangent bun- dle T ∗B over a CROSS (B,g). We denote canonical coordinates on T ∗B by (q,p) and denote by g∗ the Riemannian metric on T ∗B in- duced by g. For r > 0 we abbreviate ∗ ∗ Tr B = {(q,p) ∈ T B ||p| ≤ r} .
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