Six questions, a proposition and two pictures on Hofer distance for hamiltonian diffeomorphisms on surfaces

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1 1 Six questions, a proposition and two pictures on Hofer distance for hamiltonian diffeomorphisms on surfaces Frédéric Le Roux Laboratoire de mathématiques CNRS UMR 8628 Université Paris-Sud, Bat Orsay Cedex FRANCE September 2, 2007 This notes consists only in easy remarks, examples and questions concerning Hofer distance, that appeared at the problems session of the Snowbird Conference 1. We consider a compact surface Σ which is either the unit disc in the plane, the 2-sphere, or the closed annulus S 1 [0, 1]. Each one is equipped with its canonical area (symplectic) form. We are concerned with the groups Homeo(D 2, D 2, ω), Homeo 0 (S 2, ω), Homeo(A, A, ω, 0). The first one is the group of area preserving homeomorphisms of the disc which are the identity near the boundary; the second one is the group of area preserving homeomorphisms of the sphere isotopic to the identity; the last one is the group of area preserving homeomorphisms of the annulus which are the identity near the boundary, isotopic to the identity, and have zero mean rotation number. Remember the following open question ([2]): Question -1. Are these groups simple? Müller and Oh have defined in each context a normal subgroup Hameo whose elements are called hameomorphisms, leaving unsolved the following question ([6]). Question 0. For each of the three homeomorphisms groups above, is Hameo a proper subgroup? Let us recall the definition of Hameo, following [6]. For each of the three homeomorphisms groups we consider the subgroup of smooth diffeomorphisms, which are called Hamiltonian diffeomorphisms: they are the time one maps of timedependent compactly supported Hamiltonian flows. These subgroups, which we denote by Ham(Σ), are equipped with the Hofer metrics, defined as follows. Let H : (t, x) H t (x) be a smooth compactly supported 2 time-dependent function on Σ = D 2, S 2 or A. Then the Hofer norm of H is H := 1 0 Osc(H t )dt, where Osc(H t ) := max H t min H t. Σ Σ 1 Symplectic Topology and Measure-Preserving Dynamical Systems, Snowbird, July That is, there exists a neighbourhood of Σ on which H t = 0 for every t.

2 2 For any Hamiltonian diffeomorphism Φ, we define ρ(φ) as the infimum of the norms of the hamiltonian functions whose time one map Φ 1 H is equal to Φ. Then the Hofer distance is the bi-invariant distance defined by d(φ, Ψ) := ρ(ψφ 1 ). The difficult part is to prove non-degeneracy (see for example [5, 7]). Alternately, one will get the same distance by using the sup x,t -norm for H (see [8]). To get some feeling about this distance, remember that for any fixed t, for any arc γ in Σ with end-points x, y, the value of H t (x) H t (y) is the (algebraic) flux, through the arc γ, of the divergence-free vector field associated to H t. Thus a diffeomorphism is not too far from the identity with respect to the Hofer distance if points can be continuously moved to the position prescribed by Φ so that at each time of the move, the algebraic flux through any arc is not too big: there cannot be a big region of points moving too fast in the same direction. Now a homeomorphism h in any of our three groups is said to be a hameomorphism if there exists a continuous isotopy (h t ) from the identity to h, called a hamiltonian isotopy, and a sequence (H i ) i 0 of hamiltonian time-dependent functions on Σ, such that 1. the sequence (H i ) is a Cauchy sequence for the Hofer norm (so that it converges to a function H with 1 0 Osc(H t)dt < + ); 2. the sequence of smooth isotopies generated by H i converges to the isotopy (h t ) for the C 0 -distance given by d((h t ), (h t)) = sup t,x d Σ (h t (x), h t(x)). Furthermore, all isotopies are required to have a common compact support in the interior of Σ. Trying to answer question 0 raises further questions about the Hofer distance. In that respect, it is known that the Hofer-diameters of the groups Ham(Σ) are infinite. On the disc this is due to the existence of the Calabi invariant 3. On the sphere, this is proved by Polterovich ([8]). On the annulus, Py even shows that there exists flat subspaces of any dimension ([9], see also below). In spite of these results the global properties of the Hofer distance remains mysterious. 1 Some questions 1.1 Topological aspects of Hofer distance Question 1. Can one find h Homeo(D 2, D 2, ω) such that for any sequence (Φ i ) Ham(D 2 ) converging to h in the C 0 topology, the Hofer distance ρ(φ i ) tends to +? The same question holds on the sphere or the annulus. In other words, we would like to find a homeomorphism that is infinitely far away from diffeomorphisms. A positive answer would at once imply a positive answer to question 0. The problem is that we know no topological property that is stable and that gives a big lower bound on Hofer metric. For instance, on the 2-disc, the Calabi invariant of some hamiltonian diffeomorphism Φ gives a lower bound on ρ(φ), but diffeomorphisms with zero Calabi invariant are easily seen to be C 0 -dense in Diffeo(D 2, D 2, ω). An easier but still unsolved problem is the following. 3 The Calabi invariant of Φ is defined as the integral over time and space of any hamiltonian function H whose time one map is Φ; this quantity does not depend on H (see [1], and [3, 4] for the alternative geometric definition as an average asymptotic linking number).

3 3 Question 2. For M > 0, define the set E M := {Φ Ham(D 2 ), ρ(φ) > M}. Can one prove that for every M > 0, the set E M has non-empty interior with respect to the C 0 topology? It is well-known that Hofer distance is not continuous in the C 0 -topology: the identity map can be C 0 -approximated by a diffeomorphism with arbitrarily big Calabi invariant. But it might turn out to be semi-continuous. Question 3. Let (Φ i ) be a sequence in Ham(Σ) that C 0 -converges to Ham(Σ). Does the following inequality hold: ρ(φ) lim inf(ρ(φ i ))? A positive answer would imply a positive answer to question 2. We next ask the question of the topological invariance of the Hofer distance. Question 4. Let Φ 1, Φ 2 Ham(D 2 ), let h Homeo(D 2, D 2, ω), and assume Φ 2 = hφ 1 h 1. Does the equality ρ(φ 1 ) = ρ(φ 2 ) hold? The semi-continuity of Hofer distance (question 3) would imply a positive answer: approximate the homeomorphism h by an area preserving diffeomorphism, use the invariance of Hofer distance under conjugacy by such a diffeomorphism, and conclude by semi-continuity. 1.2 Hofer distance and curves For the next question, we turn to the group Ham(A). If x is a fixed point of Φ Ham(A), the translation number of x is defined as the number of turns made by x under an isotopy from the identity to Φ : more precisely, choose an isotopy (Φ t ) from the identity to Φ among compactly supported homeomorphisms, and consider the loop t θ(φ t (x)) where θ is the projection A S 1 ; then the translation number of x is the degree of this loop. This number does not depend on the choice of the isotopy. Alternatively, one can take any arc α joining a point of the boundary of A to x, concatenate Φ(α) with the arc α with the reverse orientation, and take the degree of the projection of this loop on S 1. Now note that the Hofer diameter of Ham(A) is infinite. Indeed, consider Φ Ham(A) with the following property: there is a smoothly embedded annulus A Int(A) = S 1 (0, 1) of area 1/2 all points of which are fixed under Φ, with translation number n. Then the Hofer distance satisfies ρ(φ) n area(a ) = n/2. Indeed, let H be a Hamiltonian function generating an isotopy (Φ t H ) from the identity to Φ. Consider the universal covering π : Ã A, and let Φ be the lift of Φ obtained by lifting the isotopy Φ t H. Choose a rectangle R made of n successive copies of a fundamental domain of A : the hypothesis on Φ entails that R is displaced by Φ. By truncating the function H π near infinity, we get a compactly supported Hamiltonian function H on Ã, generating an isotopy (Φt H e ), satisfying: 1. for every t, Osc( H t ) = Osc(H t ), and in particular H = H ; 2. Φ 1 e H = Φ on R.

4 4 Thus R is also displaced by Φ 1 e H. Now the energy-capacity inequality (applied in R 2 Ã, see for example [5, 7]) yields H = H area(r) = n area(a ). The following question is an attempt to replace the annulus A by a disc. Question 5. Let D be a smooth closed disc in A which is non displaceable (that is, Area(D) > 1/2Area(A)). Let n > 0, and Φ n be a hamiltonian diffeomorphism of A such that every point of D is a fixed point of Φ n with translation number n. Does the sequence (ρ(φ n )) tends to +? Does it holds that (ρ(φ n ) n area(d)? In case the answers are negative, one could replace the disc D with a sequence D n whose area converges to the area of the annulus A. Note that if the disc D is displaceable, then for every n > 0 there exists Φ fixing every points of D with translation number n, with ρ(φ) 2 (choose Ψ such that Ψ(D) D = with ρ(ψ) 1, and define Φ = Ψ Ψ where Ψ is a diffeomorphism adequately conjugate to Ψ that sends Ψ(D) to D). Coming back to the disc, we consider the set C of smooth loops γ : S 1 Int(D 2 ) such that γ(0) = (0, 0). Définition 1.1. To each curve γ C we associate a number l(γ), called the Hofer length of γ, defined by: l(γ) := inf { H, H H and t [0, 1], Φ t H(0) = γ(t) }. Here H denotes the space of Hamiltonian loops, that is, smooth functions H : D 2 S 1 R generating an isotopy (Φ t H ) such that Φ1 H = Id. Note that if we were not restricting to hamiltonian loops, then we would get l(γ) = 0 for every γ: in other words, every smooth curve can be realised as the trajectory of some points under a Hamiltonian isotopy with arbitrarily small Hofer length (displacing one point costs nothing). Similarly, every γ C is the trajectory of 0 under a hamiltonian loop (Φ t H ) entirely included in an arbitrarily small neighbourhood of the Identity map for the Hofer distance; but this does not imply that the loop is short. Indeed, in the next section we will show that the function l is unbounded on C. Question 6. Evaluate the Hofer length l(γ) in terms of the geometry of γ. Is it related (equivalent?) to the area surrounded by γ? By the area surrounded we mean either of the following notions. The algebraic area surrounded by γ C is the number defined by the two (coinciding) formulae λ = γ deg(x, γ)dω(x) D 2 where λ is any one-form such that dλ = ω, and for every point x that does not belong to the image of γ, the number deg(x, γ) is the number of turns made by γ around x. The absolute area surrounded by γ is the number D 2 deg(x, γ) dω(x). (...) 4 4 (...) Test : n tours d un disque. Si le disque est déplaçable, borné? Sinon, on tombe sur la question précédente?

5 5 2 Hofer-long closed curves 2.1 Construction Proposition 2.1. There exist smooth curves γ C in the unit disc with arbitrarily big Hofer length. Such a loop will be realised as γ : t Φ t H (0) where Φ1 H is a C1 approximation of a wild hameomorphism f. Proof. Consider f Homeo(D 2, D 2, ω) with the following properties: 1. the point 0 = (0, 0) is fixed under f, 2. near 0 f is a fibered rotation: it acts on the circle centered at 0 with radius r as a rotation of rotation number ρ(r), 3. the function r rρ(r) tends to + when r tends to 0, and is not integrable. Lemma. Let (H i ) be a sequence of hamiltonian functions generating isotopies whose time one maps (Φ i ) C 0 -converges to f. 1. Suppose that for every i, for every t [0, 1], Φ t H i (0) = 0. Then the sequence of Hofer lengths ( H i ) tends to The same conclusion holds if we suppose more generally that the curves γ i : t Φ t H i (0) are smooth closed curves whose sequence of Hofer lengths is bounded. Let us prove the lemma. For the first item, we see D 2 as the unit disc in R 2 and extend fby the identity outside D 2. Consider the universal covering M of M = R 2 \ {0}, equipped with the lifted area form. Let f be the lift of f R 2 \{0} that is the identity on the lift of D 2. Thanks to hypothesis 3 on f, we can find a closed topological disc R in M, which projects to a small annulus around 0 in D 2, which has arbitrarily big area in M, and such that f(r) R =. Since 0 is fixed under Φ t (hypothesis of the first item), we may lift the isotopy (Φ t H i ) to an isotopy ( Φ t H i ) of M. The maps ( Φ 1 H i ) C 0 -converges to f uniformly on compact subsets of M, so that for i big enough the domain R is still displaced by Φ 1 H i. Using a variation of the argument used to show that the Hofer diameter of Ham(A) is infinite (before question 5), we conclude that the Hofer norm H i is arbitrarily big. For the second item, the hypothesis gives us a sequence of Hamiltonian loops (K i ) in H with bounded Hofer norms, and such that for every i, Φ t K i (0) = Φ t H i (0). Let Ψ i := (Φ t K i ) 1 Φ t H i. This defines a new sequence that satisfies the hypotheses of the first item, Thus its Hofer norms tends to infinity, and it follows that the sequence ( H i ) also tends to infinity. This proves the lemma. We now turn to the construction of a sequence of loops in D 2 with Hofer lengths tending to infinity. Let f be a hameomorphism satisfying properties 1,2,3 as above: such an f we may obtain by taking any g Homeo(D 2, D 2, ω) that satisfies the same properties and supported on a small disc D, any hamiltonian diffeomorphism h displacing D, and letting f = [g, h] = ghg 1 h 1. As the hameomorphisms constitute a normal subgroup containing the hamiltonian diffeomorphisms, f is indeed a hameomorphism. In particular there exists a sequence (H i ) of hamiltonian functions with bounded Hofer norms such that (Φ 1 H i )

6 6 C 0 -converges to f. Furthermore, since (Φ 1 H i (0)) tends to 0, one can modify the sequence H i, keeping the previous properties, so that γ i : t Φ t H i (0) is a closed smooth curve, i. e. it belongs to C. Alternatively, one can give a more explicit construction. Choose g which is smooth except at 0, approximate g by a hamiltonian diffeomorphism g i which is again a fibered rotation supported on D, and is equal to g except on a small neighbourhood of 0. Now take Φ t H i = [g i, h t ], where h t = Φ t L where the hamiltonian function L t is everywhere zero for t near 0 and 1. Then the Hofer norm of H i is less than 2 L for every i, and furthermore the curve γ i is closed and smooth (with a stationary point at t = 0, 1). If the sequence of lengths of (γ i ) was bounded, then the Hofer norms of (H i ) would tend to infinity (apply item 2 of the lemma). Thus we see that the sequence (γ i ) has unbounded Hofer length. This completes the proof of the proposition. Note that the sequence of isotopy (Φ t H i ) C 0 -converge to the continuous isotopy f t = [g, h t ]. Thus the sequence of curves (γ i ) converges to a continuous curve γ : t f t (0). 2.2 Picture of a Hofer-long curve Figure 1 shows a numerical simulation of the curve γ in the proof of the proposition, for the homeomorphism g generated by the singular Hamiltonian H(θ, r) = 2πr( 1 1). The map g is a fibered rotation, and rotation number on r 2 the circle of radius r is ρ(r) = 1 1. More explicitely, the curve γ is given by the r 2 formula γ(t) = gh t g 1 h t (0), where h t (x, y) = (x + t, y). The curve is smooth except at the (infinitely many) values of the parameter t = 1 n, n =..., 3, 2, 1 for which γ(t) = 0. Between two such successive values, γ(t) runs on a simple closed curve with a single non-smooth point (an infinitely twisted circle). The first picture gives the last loop of the curve, corresponding to t [ 1 2, 1] (the part near 0 is missing, the whole central disc should be black). The second figure depicts the whole curve (here the segments near 0 are of course artefacts of the numerical computation). Note that here the support of f exceeds the unit disc (g should first be truncated, but this would not change the general aspect of the picture). Also note that g can be approximated by a diffeomorphism that coincides with it except on some small neighbourhood of 0 ; this gives a smooth approximation γ i of the curve γ that coincides with it except on a small neighbourhood of 0, and with arbitrarily big Hofer length. An animation displaying the image of a segment of the x-axis under the isotopy can be found on my webpage 5. It shows that points move very fast (which is unavoidable for an isotopy from the identity to f, since at time one lots of couples of points will have made lots of turns around each other) but with no big region having fast coherent behaviour, as is necessary for a C 0 -hamiltonian isotopy as defined by Müller and Oh. 3 Two more remarks As a conclusion, let us mention two variations on Müller-Oh definition of hameomorphisms of D 2. First, one can require that the function H i,t has zero mean for 5 leroux/

7 7 1,2 0,8 0,4-2 -1,6-1,2-0,8-0,4 0 0,4 0,8 1,2 1,6 2-0,4-0,8-1,2 1,2 0,8 0,4-2 -1,6-1,2-0,8-0,4 0 0,4 0,8 1,2 1,6 2-0,4-0,8-1,2 Figure 1: Motion of one point under a C 0 -hamiltonian isotopy, joining the identity to the hameomorphism f

8 8 every i and t. Thus one gets a normal subgroup of Homeo(D 2, D 2, ω) which is a subgroup of Hameo (heuristically, this should be the subgroup of hameomorphisms with vanishing Calabi invariant). The second variation consists in adding the condition Φ t H i (0) = 0 for every i, t. Thus we get a group of homeomorphisms which is normal in the subgroup of elements of Homeo(D 2, D 2, ω) fixing 0. By identifying homeomorphisms which coincide near 0, one gets a normal subgroup of the group of germs of area and orientation preserving homeomorphisms of the plane. In this context it can be proved that this is a proper subgroup (the argument amounts to item 1 of the above lemma). Of course, this is not the easiest way to prove that this group is not simple (consider the subgroup of elements whose fixed point set has density one near 0). References [1] Arnol d, V. I. and Khesin, B. Topological Methods in Hydrodynamics Springer, [2] Fathi, A. Structure of the group of homeomorphisms preserving a good measure on a compact manifold. Annales scientifiques de l École Normale Supérieure Sér. 4, 13 (1980) no. 1, [3] Fathi, A. Transformations et homéomorphismes préservant la mesure. Systemes dynamiques minimaux. Thèse, Orsay (1980). [4] Gambaudo, J.-M. et Ghys, E. Enlacements asymptotiques. Topology 36 (1997), no. 6, [5] Hofer et E. Zehnder Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, [6] Oh, Y.-G. and Müller, S. The group of hamiltonian homeomorphisms and C 0 -symplectic topology. arxiv:math/ , [7] Polterovich, L. The Geometry of the Group of Symplectic Diffeomorphisms. Birkhäuser, [8] Polterovich, L. Hofer s diameter and Lagrangian intersections International Mathematics Research Notices (1998), Issue 4, Pages [9] Py, P. Quelques plats pour la métrique de Hofer. Prepublication, avril 2007.