On primes and period growth for Hamiltonian diffeomorphisms

Here we use Vinogradov's prime distribution theorem and a multi-dimensional generalization due to Harman to strengthen some recent results concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we partially recover, using only symplectic tools, a theorem on the periodic points of Hamiltonian diffeomorphisms of the sphere by Franks and Handel.


Introduction and main results
1.1. Introduction. Hamiltonian diffeomorphisms tend to have infinitely many periodic points as well as periodic points with arbitrarily large periods. For Hamiltonian diffeomorphisms of the two-dimensional sphere these tendencies are illustrated by the following two beautiful theorems by Franks, and Franks and Handel. Theorem 1.1. ( [F1, F2]) If φ : S 2 → S 2 is a Hamiltonian diffeomorphism then φ has either two or infinitely many periodic points.
Theorem 1.2. (Theorem 1.4 of [FH]) If φ : S 2 → S 2 is a nontrivial Hamiltonian diffeomorphism with at least three fixed points, then there exists integers n > 0 and K > 0 such that φ n has a periodic point of period k for every k ≥ K.
. It is expected that results similar to Theorems 1.1 and 1.2 hold for the Hamiltonian diffeomorphisms of symplectic manifolds, such as CP n , with positive first Chern number. With such generalizations in mind, a new proof of Theorem 1.1 which uses only tools from symplectic topology was established in [CKRTZ]. The initial goal of this paper was to extend these methods to obtain a symplectic proof of Theorem 1.2. Here we fall short of this goal. Most significantly, it is not clear to the author how to deal with nonisolated fixed points using the methods of [CKRTZ], and so we need to assume more than the simple hypothesis of nontriviality from Theorem [FH]. Even with a stronger hypothesis we do not recover the linear period growth of 1.2. Instead we prove the following result.
Date: May 1, 2014. 2000 Mathematics Subject Classification. 53D40, 37J45, 70H12. This work was partially supported by a grant from the Simons Foundation (207839). The author also acknowledges support from National Science Foundation grant DMS 08-38434 EMSW21-MCTP: Research Experience for Graduate Students. Theorem 1.3. Let φ be a Hamiltonian diffeomorphism of S 2 with at least three fixed points such that Fix(φ k ) is finite for all k ∈ N. Then φ has fixed points of arbitrarily large period. More precisely, either φ has infinitely many periodic points with arbitrarily large period and rational mean indices not equal to zero modulo four, or there exists a K ∈ N such that φ K has infinitely many periodic points with arbitrarily large periods in P 2 , the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity.
Two new results of some independent interest and more general scope arise from the proof of Theorem 1.3. Both of these are established using deep theorems from number theory concerning the distribution of prime numbers. More precisely, using Vinogradov's prime distribution theorem (Theorem 2.1 below), we prove the following sharper version of Theorem 1.18 (part (i)) from [GG] where the existence of infinitely many contractible periodic points is established under the same hypotheses.
Theorem 1.4. Suppose that (M, ω) is a closed, weakly-monotone and rational symplectic manifold. Let ϕ be a Hamiltonian diffeomorphism of (M, ω) which has finitely many contractible fixed points one of which is a symplectically degenerate maximum. Then ϕ has infinitely many distinct contractible periodic points of arbitrarily large prime period.
Using Theorem 1.4 together with a multidimension generalization of Vinogradov's theorem due to Harman (Theorem 2.2 below), we then augment a result from [GK] to establish the existence of resonance relations for the fixed points of certain Hamiltonian diffeomorphisms.
Theorem 1.5. Suppose that (M, ω) is a closed, weakly-monotone and rational symplectic manifolds whose minimal Chern number N is greater than half its dimension. Let ϕ be a Hamiltonian diffeomorphism of (M, ω) which has finitely many contractible fixed points and does not have contractible periodic points of arbitrarily large period in P 2 . Then the nonzero mean indices of the contractible fixed points of ϕ satisfy at least one nontrivial resonance relation (i.e., they satisfy a nontrivial linear equation with integer coefficients which is homogeneous modulo 2N ).
The relevant theorem from [GK] establishes identical restrictions on the periodic point set of Hamiltonian diffeomorphisms with finitely many such points. A fundamental example covered by Theorem 1.5 and not by [GK] is that of a nontrivial rotation of S 2 by a rational multiple of 2π.
The effectiveness of these number theoretic results on the distributions of primes in extending the theorems from [GG] and [GK] is the consequence of two factors. The first of these is the simple fact that when considering the growth of periods for the periodic points of a map φ the following alternative is often useful; a prime iterate of φ, say φ p , either has the same fixed point set as φ, or φ has a periodic point of period p. The other factor has to do with a common feature of the proofs of the theorems from [GG, GK]. For a Hamiltonian diffeomorphism of a rational symplectic manifold (M, ω) whose minimal Chern number N is finite, the actions of contractible fixed points are defined modulo the index of rationality, and their mean indices are defined modulo 2N . When there are finitely many such fixed points, the collection of their actions or mean indices can then be viewed as points on a torus. Since the action and mean index of a fixed point both grow linearly under iteration, under suitable assumptions, the collections of actions and mean indices follow linear paths on a fixed torus as the map is iterated. This scenario occurs in both [GG] and [GK] where restrictions on these linear paths, coming from the insights at the hearts of the proofs of the Conley conjecture from [Hi, Gi], are then used to infer the desired conclusions. The results of Vinogradov and Harman mentioned above allow us to detect the same restrictions on these linear paths while at the same time exploiting the advantage of considering only prime iterations.
1.2. Organization. In the next section we recall the theorems from number theory and the symplectic tools needed in the rest of the paper. The proofs of Theorem 1.4 and Theorem 1.5 are contained in Sections 3 and 4, respectively. Theorem 1.3 is then proved in Section 5.
1.3. Acknowledgments. The author thanks his colleagues Scott Ahlgren and Alexandru Zaharescu for answering his naive number theory questions. He would also like to thank Viktor Ginzburg for useful discussions.

Preliminaries
2.1. Number theoretic tools. Let p j denote the jth prime number. By convention, we will set p 0 = 1 although we will refrain from calling 1 a prime number. Theorem 2.1. ( [Vi]) If ϑ is irrational then the sequence (p j ϑ) is uniformly distributed on (0, 1).
To prove Theorem 1.5 we will also require a multidimensional generalization of Theorem 2.1 due to G. Harman, [Ha]. In fact, the following immediate implication of Theorem 1.4 of [Ha] is sufficient for our purposes.
2.2.1. Symplectic isotopies. Let ψ t be a smooth isotopy of symplectic diffeomorphisms of (M, ω), where t takes values in [0, 1] and ψ 0 is the identity map. Denote the set of fixed points of ψ 1 by Fix(ψ 1 ), and the set of periodic points by Per(ψ 1 ). The period of a point P ∈ Per(ψ 1 ) is defined to be the smallest positive integer k for which P ∈ Fix((ψ 1 ) k ). We will assume throughout that the generating vector fields of our symplectic isotopies ψ t are all time-periodic with period one. This assumption imposes no new restrictions on the time-one maps considered as any symplectic isotopy is homotopic, relative its endpoints, to one with this property. It allows us to consider the path ψ t for all t ∈ R and to identify ψ k with (ψ 1 ) k .
The subset of symplectic isotopies we are most interested in are Hamiltonian flows. A Hamiltonian on (M, ω) is a function H : R/Z × M → R, or equivalently a smooth one-periodic family of functions H t (·) = H(t, ·). Each such Hamiltonian determines a one-periodic vector field X H on M , via the equation i XH ω = −dH t , whose time-t flow will be denoted by φ t H . This flow is defined for all t ∈ R and defines a smooth isotopy of symplectic diffeomorphisms. The set of time one maps φ = φ 1 H of Hamiltonian flows constitutes the group Hamiltonian diffeomorphisms of (M, ω).

The Conley-Zehnder and mean indices.
Consider a smooth isotopy ψ t of symplectic diffeomorphisms as above. For P ∈ Fix(ψ 1 ), let x : [0, 1] → M be the closed curve ψ t (P ). Given a symplectic trivialization ξ of x * T M , the linearized flow of ψ t along x(t) yields a smooth path A ξ : [0, 1] → Sp(n) of symplectic matrices starting at the identity matrix. One can associate to A ξ its Conley-Zehnder index µ(A ξ ) ∈ Z as defined in [CZ], and its mean index ∆(A ξ ) ∈ R as defined in [SZ]. These quantities depend only on the homotopy class of the symplectic trivialization ξ. We denote this class by [ξ] and define the Conley-Zehnder and mean index of P with respect to this choice as respectively. The following properties of these indices are proved in [SZ].
Approximation. The mean index is never too far from the Conley-Zehnder index as where the strict form of the inequality holds if the linearization of ψ 1 at P has at least one eigenvalue different from 1.
Iteration formula. The mean index grows linearly under iteration, i.e., where ξ k is the trivialization of T M along ψ tk (X) induced by ξ.
Continuity. The mean index is also continuous with respect to C 1 -small perturbations of the symplectic isotopy, [SZ]. To state this property precisely, we first note that if two fixed points P and P ′ , of possibly different maps ψ 1 and ψ ′ 1 , represent the same homotopy class c ∈ π 1 (M ) then any choice of [ξ] for P determines a unique class of symplectic trivializations for P ′ which we still denote by [ξ]. When we compare indices of fixed points in the same homology class we will always assume that the classes of trivializations being used are coupled in this manner. Now let ψ t be a symplectic isotopy C 1 -close ψ t . Under this perturbation, each fixed point P of ψ 1 splits into a collection of fixed points of ψ 1 which are close to P (and hence in the same homotopy class as P ). If P is one of these fixed points of ψ 1 then Finally we recall that in dimension two there is, generically, a very simple relation between the mean and Conley-Zehnder indices. The following result can be easily derived, for example, from Theorem 7 in Chapter 8 of [Lo].
Lemma 2.3. Let (M, ω) be a two-dimensional symplectic manifold and suppose that ψ t is an isotopy of symplectic diffeomorphisms of (M, ω) starting at the identity. If P is a fixed point of ψ 1 and ∆(P ; ψ t , [ξ]) is not an integer, then µ(P ; ψ t , [ξ]) is the odd integer closest to ∆(P ; ψ t , [ξ]).

2.2.3.
Indices of contractible fixed points modulo 2N . When P is a contractible fixed point of ψ t , that is x(t) = ψ t (P ) is contractible, it is often useful to restrict attention to trivializations of x * T M determined by a choice of smooth spanning disc u : D 2 → M with u(e 2πit ) = x(t). For such choices of trivializations the corresponding indices are well-defined modulo twice the minimal Chern number, 2N . In fact, the corresponding elements of R/2N Z depend only on the time one map ψ 1 and hence will be denoted by µ(P ) and ∆(P ).

2.2.4.
Actions of contractible fixed points modulo λ 0 . Let P be a contractible fixed point of a Hamiltonian diffeomorphism φ 1 H and set x(t) = φ t H (P ). If u : D 2 → M is a spanning disc for x(t) as above, the action of P with respect to u is defined to be This is what we will call the action of P with respect to H. The action spectrum of H is defined here to be the set The action, like the mean index grows linearly with iteration, in the sense that for all k ∈ N and P ∈ Fix(φ 1 2.2.5. Hamiltonian Floer homology. Let H be a Hamiltonian on (M, ω). The contractible 1-periodic orbits of of X H are in one-to-one correspondence with the contractible fixed points of φ 1 H . Assuming these are all nondegenerate they generate the Floer complex of H and the corresponding Hamiltonian Floer homology of H, HF(H). This is isomorphic to HQ(M ), the quantum homology of M . More precisely, with respect to the standard gradings of HF(H) by the Conley-Zehnder index, we have HF * (H) ∼ = HQ * +n (M ).
We will also require two related notions. The first of these, introduced by Floer and Hofer in [FH], is HF (a,b) (H), the Hamiltonian Floer homology of H restricted to the action window (a, b). The second required variant of Floer homology needed is that of local Floer homology as introduced by Floer in [Fl]. In the present setting, the local Floer homology is a group HF loc * (H, P, u) associated to an isolated contractible fixed point P of φ 1 H and a spanning disc u : D 2 → M for the smooth loop x(t) = φ t H (P ). The only detail of the construction of local Floer homology relevant here is that the first step is to perturb H, if necessary, so that P breaks into a collection nearby fixed points which are nondegenerate.
2.2.6. Floer homology of symplectic diffeomorphisms. Finally, we will need to consider the Floer homology of symplectic diffeomorphisms. In fact, we need only consider the very special case when (M, ω) is a two-dimensional symplectic torus (T 2 , Ω) and the symplectic diffeomorphism is isotopic to the identity. The relevant details of this construction are described briefly below and the reader is referred to [DS,Se1,Se2,Se3,LO] for more details on the general construction, and to [C1, C2] for more thorough reviews of the Floer theory of symplectic diffeomorphisms of surfaces.
Consider a smooth isotopy ψ t of symplectic diffeomorphisms of (T 2 , Ω) starting at the identity such that the fixed points of ψ 1 are all nondegenerate. The Floer homology of ψ 1 , FH(ψ 1 ), is then a well-defined group with the following properties.
Splitting: The Floer homology FH(ψ 1 ) admits a decomposition of the form Here, each summand FH(ψ 1 ; c) is the homology of a chain complex (CF(ψ 1 ; c), ∂ J ) where the chain group CF(ψ 1 ; c) is a torsion-free module over a suitable Novikov ring, and the rank of this module is the number of fixed points of ψ 1 which represent the class c. The group FH(ψ 1 ; 0) coincides with the Floer-Novikov Homology constructed by Lê and Ono in [LO], ([Se3]). Moreover, if ψ t is homotopic to a Hamiltonian isotopy φ t H , then FH(ψ 1 ) = FH(ψ 1 ; 0) = HF(H), [Fl]. In particular, FH(ψ 1 ) is canonically isomorphic to H(T 2 ; Z).
Grading: Each chain complex (CF(ψ 1 ; c), ∂ J ) above has a relative Z-grading and the boundary operator decreases degrees by one. For the case c = 0, the grading can be set by using the usual Conley-Zehnder index of contractible fixed points (which is well-defined since c 1 (T 2 ) = 0.) For example if ψ t is homotopic to a Hamiltonian isotopy we have FH * (ψ 1 ; 0) = H * +1 (T 2 ; Z). (2.8) For a general class c ∈ H 1 (T 2 ; Z) the (relative) grading of (CF * (ψ 1 ; c), ∂ J ) is again determined by the Conley-Zehnder index and the overall shift can be fixed by choosing a homotopy class of symplectic trivializations of z * (T T 2 ) where z : S 1 → T 2 is a smooth representative of c.
Extension to all smooth isotopies: The property of invariance under Hamiltonian isotopy allows one to also define the Floer homology for any smooth symplectic isotopy ψ t of (T 2 , Ω). One simply sets where φ is a Hamiltonian diffeomorphism for which the fixed points of ψ 1 • φ are nondegenerate. For example, if ψ t = id for all t ∈ [0, 1] we can perturb by the Hamiltonian flow of a C 2 -small Morse function to obtain FH(id) = FH(id; 0) = H(T 2 ; Z). (2.9) Dichotomy: The following alternative is well known and plays a crucial role in the proof of Theorem 1.3. A proof of this alternative can be found in [CKRTZ], for example.

Vinogradov's Hanukkah gift: Symplectically degenerate maxima and periodic orbits of prime period
Let ϕ = φ 1 H be a Hamiltonain diffeomorphism of (M, ω) and let P be a fixed point of ϕ such that the loop x(t) = φ t H (P ) is contractible. The point P is called a symplectically degenerate maximum of ϕ if for some spanning disc u : D → M of  [Hi] and Ginzburg in [Gi]. It is established in this more general setting by Ginzburg and Gürel in [GG].
Theorem 3.1. (Theorem 1.17 of [GG]) Suppose (M, ω) is weakly-monotone and rational and that P is a symplectically degenerate maximum of ϕ = φ 1 H . Let u be a spanning disc as in (3.1) and set c = A H (P, u). For every sufficiently small ǫ > 0 there is a k ǫ > 0 such that for all k > k ǫ and some δ k with 0 < δ k < ǫ.
Also proved in [GG] is the following result concerning the periodic points of a Hamiltonian diffeomorphism with a symplectically degenerate maximum.
Theorem 3.2. (part (i) of Theorem 1.18 of [GG]) Suppose (M, ω) is weaklymonotone and rational. If ϕ has finitely many contractible fixed points and one of these is a symplectically degenerate maximum, then ϕ has infinitely many geometricially distinct contractible periodic points.
In this section we prove Theorem 1.4 which improves Theorem 3.2 by detecting, under the same hypotheses, infinitely many contractible periodic points of arbitrarily large prime period.
3.1. Proof of Theorem 1.4. In fact, we only need to alter the proof from [GG] by considering prime iterates and replacing the role of the standard equidistribution theorem with Vinogradov's prime distribution theorem. Because it is short, we include the entire argument for completeness, following very closely the presentation from [GG].
Arguing by contradiction we assume that ϕ = φ 1 H does not have contractible periodic points of arbitrarily large prime period and derive a contradiction. Let P be a symplectically degenerate maximum of φ 1 H and u a spanning disc of φ t H (P ) such that (3.1) holds. For convenience, we will normalize (add the appropriate constant to) our generating Hamiltonian H so that A H (P, u) = 0.
If φ 1 H does not have periodic points of prime period set j 0 = 0. Otherwise, let j 0 > 0 be the smallest natural number such that p j0 is greater than the largest prime period of φ 1 H . We then have for all j ≥ j 0 . Now, the action spectrum of H, S(H) can be divided into action values which are rational and irrational modulo the index of rationality, λ 0 , i.e., where the η i are irrational and the m i and n i are integers. From (3.3) we also have for all j ≥ j 0 where, again, the flow of the Hamiltonian H pj is φ tpj H . Let d be the least common multiple of the n i . Choose the ǫ in Theorem 3.1 to be less than 1/d and let k ǫ be the corresponding integer promised in that theorem. Choose j 1 ≥ j 0 such that p j > k ǫ for all j ≥ j 1 . Theorem 3.1 implies that for all primes p j > k ǫ one of the elements of S(H pj ) lies in the interval (0, ǫ) ⊂ R/λ 0 Z. For all j ≥ j 1 it follows from (3.5) and our choice of ǫ that this element of S(H pj ) in (0, ǫ) must be of the form p j η i for some i = 1, . . . , r. In conclusion, under our assumption on ϕ, for all sufficiently small ǫ > 0 there is a j 1 ≥ j 0 such that for every j ≥ j 1 there is an integer i ∈ [1, r] such that p j η i ∈ (0, ǫ). We now derive the desired contradiction by showing that this statement is false.

Resonance relations for fixed points of Hamiltonian diffeomorphisms
Let ϕ be a Hamiltonian diffeomorphism of (M, ω) which has finitely many contractible fixed points. Let ∆ 1 , . . . , ∆ m be the collection of nonzero mean indices of the contractible fixed points of ϕ. A resonance relation for Fix(ϕ) is a vector a = (a 1 , . . . , a m ) ∈ Z m such that The resonance relations of Fix(ϕ) form a free abelian group R(ϕ) ⊂ Z m .
In this section we use Theorem 1.4 and Theorem 2.2 to prove Theorem 1.5, which we restate here for convenience.
Theorem 4.1. Suppose that (M, ω) is weakly-monotone and rational and the minimial Chern number N is finite and greater than n. Let ϕ be a Hamiltonian diffeomorphism which has finitely many contractible fixed points and does not have contractible fixed points of arbitrarily large period in P 2 . Then R(ϕ) = 0, i.e., the nonzero mean indices ∆ i satisfy at least one non-trivial resonance relation.
Again P 2 is the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. Proposition 4.2. There is a j 0 ≥ 0 such that for any j ≥ j 0 the point p j ∆(ϕ) = (p j ∆ 1 , . . . , p j ∆ m ) ∈ (R/2N Z) m is not contained in Π.
Proof. If ϕ does not have periodic points with period in P 2 set j 0 = 0. Otherwise, let j 0 > 0 be the smallest natural number such that p j0 is greater than the largest period of ϕ in P 2 . We then have Fix(ϕ pj ) = Fix(ϕ) for all j ≥ j 0 . (4.1) Choose a Hamiltonian H with ϕ = φ 1 H . The fixed point set and the mean indices (modulo 2N ) are independent of this choice. For any Hamiltonian G on (M, ω), the Hamiltonian Floer homology, HF * (G), is graded modulo 2N by the Conley-Zehnder index and is isomorphic to HQ n+ * (M ). Since N > n, the nontrivial fundamental class HF n (G) is distinguished by its degree. In particular, for all p j we have HF n (H pj ) = 0. (4.2) For this to hold there must be a contractible fixed point X of φ 1 Hp j and a spanning disc u of φ t Hp j (X) such that the local Floer homology HF loc n (H pj , X, u) is nontrivial. For such an X it follows from the construction of local Floer homology, the continuity properties of the mean index and ( Arguing by contradiction, assume instead that X = Q i for some i ∈ [1, l]. Then Since N > n, it then follows from (4.3) that ∆(Q i ; φ t Hp j , [u]) = 0. We also have HF loc n (H pj , Q i , u) = 0. So, the point Q i is a symplectically degenerate maximum of φ 1 Hp j = ϕ pj . Theorem 1.4 then implies that ϕ pj has contractible periodic points of arbitrarily large prime period. This contradicts the hypothesis of Theorem 1.5 that ϕ does not have contractible fixed points of arbitrarily large period in P 2 .
To conclude, we note that Proposition 4.2 implies that the set is not dense in (0, 2N ) m . It then follows from Theorem 2.2 that the numbers 1, ∆ 1 , . . . , ∆ m must be rationally dependent. Hence, the rank of R(ϕ) is at least one and the proof of Theorem 1.5 is complete.

Period growth for Hamiltonian diffeomorphisms of the sphere
We now prove Theorem 1.3 which we again restate for convenience.
Theorem 5.1. Let φ be a Hamiltonian diffeomorphism of S 2 with at least three fixed points such that Fix(φ k ) is finite for all k ∈ N. Either φ has infinitely many periodic points with arbitrarily large period and rational mean indices not equal to zero modulo four, or there exists a K ∈ N such that φ K has infinitely many periodic points with arbitrarily large periods in P 2 .
5.1. Sorting and a reduction to two cases. We begin by describing a recursively defined sorting procedure. First we sort the fixed points of φ into three disjoint subsets defined in terms of their mean indices; where P is in Z 0 if ∆(P ) = 0 mod 4, P is in I 0 if ∆(P ) is irrational and P is in Q 0 otherwise. If Q 0 = ∅ we stop. Otherwise we set K 1 = min{k ∈ N | k∆(P ) = 0 mod 4 for all P ∈ Q 0 }, and then consider the map φ K1 . Sorting its fixed points into three disjoint sets as before, we get Fix(φ K1 ) = Z 1 ∪ I 1 ∪ Q 1 .
(5.2) Clearly, Fix(φ) ⊂ Fix(φ K1 ). In particular, it follows from the iteration formula for the mean index and our choice of K 1 that Z 0 ∪ Q 0 ⊂ Z 1 , I 0 ⊂ I 1 and any element of Q 1 is a periodic point of φ whose period lies in (1, K 1 ] and divides K 1 . If Q 1 = ∅ we stop otherwise we define K 2 as above, and proceed in the same manner. Either we can repeat this procedure forever or it terminates at some first step, say the N th, at which Q N is empty. In the latter case there must be a sequence of periodic points Q i ∈ Q i whose mean indices are rational and not equal to zero modulo four, such that the period of Q i is greater than i−1 j=1 K j ≥ 2 i−1 . This corresponds to the first period growth alternative of Theorem 1.3.
It remains to prove that if Q N = ∅ for some finite N then for some K ∈ N the map φ K has periodic points with arbitrarily large periods in P 2 . Arguing by contradiction, we set ϕ = φ K1···KN and assume that ϕ does not have infinitely many periodic points with arbitrarily large periods in P 2 .
By our choice of ϕ, we have where the points in Z have zero mean index modulo four, and the points in I have irrational mean indices.
Lemma 5.2. The set I has at least two elements.
Proof. By assumption, Theorem 1.5 applies to ϕ and implies that the mean indices of the elements of I must satisfy at least one nontrivial resonance relation. Since these mean indices are irrational there are at least two elements of I.
Lemma 5.3. The set Z is not empty.
Proof. Arguing by contradiction assume that Fix(ϕ) = I. Since Fix(φ) ⊂ Fix(ϕ) it follows from the hypotheses of Theorem 1.3 that there must be at least three elements of Fix(ϕ). Since they have irrational mean indices, each of these fixed points is nondegenerate and elliptic and so their topological indices are all even. By the Lefschetz fixed point theorem we then have the Euler characteristic of S 2 equal to | Fix(ϕ)| > 2, a contradiction.
Following [CKRTZ] we now separate the proof into two cases; the one in which at least one point of Z is degenerate, and the other in which every element of Z is nondegenerate. In both cases we will derive a contradiction to Proposition 2.4. 5.2. Case 1: at least one point in Z is degenerate. Assume that ϕ as above has a fixed point, say W , which is degenerate and satisfies ∆(W ) = 0 mod 4. By Lemma 5.2 we can also choose two elements X and Y of I.

5.2.1.
Step 1: A useful generating Hamiltonian for ϕ. We first choose a generating Hamiltonian H : R/Z × S 2 → R for ϕ such that W and X are fixed points of φ t H for all t ∈ R. The construction of such a Hamiltonian is described in §3.3.1 of [CKRTZ].
For each k ∈ N, the Hamiltonian diffeomorphism ϕ k is then generated by the Hamiltonian H k (t, P ) = kH(kt, P ) and φ t H k = φ kt H for all t ∈ R. So, W and X are still static fixed points of the flow of H k .

5.2.2.
Step 2: A useful perturbation. Under our assumption that ϕ does not have periodic points of arbitrarily large period in P 2 , there is a j 0 ∈ N such that Fix(ϕ p 2 j ) = Fix ϕ (5.4) for all j ≥ j 0 . For each such j we now describe a perturbation of the time-one flow of the Hamiltonian H p 2 j . Lemma 5.4. (Lemma 3.4, [CKRTZ]) For each j ≥ j 0 there is a neighborhood U j of W and a Hamiltonian flow φ j,t which is arbitrarily C ∞ -close to φ t H p 2 j , is equal to φ t H p 2 j outside of U j , and whose fixed point set has the form Fix( φ j,1 ) = Fix(ϕ) ∪ {W 1 , . . . , W d },

5.2.3.
Step 3: transfer of dynamics to the torus. We now construct from φ j,t a symplectic isotopy ψ j,t of the torus. We first blow-up the points W and X and complete the restriction of φ j,1 to S 2 {W, X} to an area preserving diffeomorphism of the closed annulis [−1, 1] × R/2πZ. The resulting map, φ j , acts on the boundary circles, Γ W = {1} × R/2πZ and Γ X = {−1} × R/2πZ, as the rotation by λ and π∆(X; φ j,t , Following Arnold (Appendix 9 of [Ar]), we now extend the map φ j to the torus formed by gluing two copies of the domain annulus [−1, 1] × R/2πZ along their common boundaries after inserting two connecting cylinders. This allows us to extend the map φ j to an area preserving map ψ j of (T 2 , Ω) which agrees with φ j on the two annuli, and is defined on the connecting cylinders so that the overall map is smooth and has no new fixed points. In particular, Fix(ψ j ) consists of two copies of Fix(φ j ), which we denote by Fix(φ ± j ) = (Z W ) ± ∪ (I X) ± ∪ {W ± 1 , . . . , W ± d }. The isotopy φ j,t induces a smooth isotopy ψ j,t from the identity to ψ j . 5.2.4. The contradiction. There are two fixed points of ψ j , Y ± , corresponding to Y . The following result concerning the role of Y + in the Floer complex generating the Floer homology FH(ψ j ) implies the desired contradiction.
Proposition 5.5. If j ≥ j 0 is sufficiently large then Y + represents a nontrivial class in FH(ψ j ), and if Y + is contractible then the degree of the class [Y + ] is greater than one in absolute value. the irrational number ∆(Y ; φ t