Symmetric Liapunov center theorem for orbit with nontrivial isotropy group
aa r X i v : . [ m a t h . C A ] O c t SYMMETRIC LIAPUNOV CENTER THEOREMFOR ORBIT WITH NONTRIVIAL ISOTROPY GROUP
MARTA KOWALCZYK , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI Abstract.
In this article we prove two versions of the Liapunov center theorem for symmetricpotentials. We consider a second order autonomous system ¨ q ( t ) = −∇ U ( q ( t )) in the presenceof symmetries of a compact Lie group Γ acting linearly on R n . We look for non-stationaryperiodic solutions of this system in a neighborhood of an orbit of critical points of the potential U. Our results generalize that of [12, 13]. As a topological tool we use an infinite-dimensionalgeneralization of the equivariant Conley index due to Izydorek, see [8]. introduction The Liapunov center theorem is one of the most significant theorems regarding the existenceof periodic solutions of ordinary differential equations in a neighborhood of stationary ones.Consider a second order autonomous system of the following form ¨ q ( t ) = −∇ U ( q ( t )) , (1.1)where the potential U : R n → R is of class C and ∈ R n is a non-degenerate critical point of U which is not a local maximum, i.e. ∇ U (0) = 0 , det ∇ U (0) = 0 and σ ( ∇ U (0)) ∩ (0 , + ∞ ) = { β , . . . , β m } for some m ≥ . Without loss of generality we assume that β > β > . . . > β m > . Now suppose that there exists β j such that β j /β j N for all j = 1 , . . . , j − . Then the famousLiapunov center theorem states that there exists a sequence ( q k ( t )) of periodic solutions of thesystem (1.1) with amplitude tending to and a sequence ( T k ) of minimal periods such that T k → π/β j as k → + ∞ , see for instance [1, 2], [7] and [11].A discussion of some generalizations of the Liapunov center theorem one can find in [12].The goal of this paper is to prove the Liapunov center theorem in the presence of symmetriesof the potential U. Therefore, from now on, we discuss symmetric versions of the Liapunov centertheorem.Let Ω ⊂ R n be an open and Γ -invariant subset of R n where R n is considered as an orthogonalrepresentation of a compact Lie group Γ . Assume that q ∈ Ω is a critical point of the Γ -invariant potential U : Ω → R of class C . Since for all γ ∈ Γ the equality U ( γq ) = U ( q ) holds and ∇ U ( q ) = 0 , the orbit Γ( q ) = { γq : γ ∈ Γ } consists of critical points of U, i.e. Γ( q ) ⊂ ( ∇ U ) − (0) . It is easy to see that dim ker ∇ U ( q ) ≥ dim Γ( q ) . The orbit Γ( q ) is callednon-degenerate if dim ker ∇ U ( q ) = dim Γ( q ) . For ε > by Γ( q ) ε we understand an ε -neighborhood of the orbit Γ( q ) , i.e. Γ( q ) ε = [ q ∈ Γ( q ) B ε ( R n , q ) , where B ε ( R n , q ) denotes the open ε -ball centered at q in R n . Date : October 23, 2018.2010
Mathematics Subject Classification.
Primary: 34C25; Secondary: 37G40.
Key words and phrases. periodic solutions, Liapunov center theorem, equivariant bifurcations, equivariantConley index. , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI We are interested in finding non-stationary periodic solutions of the system (1.1) in a neigh-borhood of the orbit Γ( q ) of stationary solutions. Note that if dim Γ ≥ then it can happenthat dim Γ( q ) ≥ , i.e. the critical point q is not isolated in ( ∇ U ) − (0) . That is why forhigher-dimensional orbits Γ( q ) one can not apply the classical Liapunov center theorem.In [12] we have proved the symmetric Liapunov center theorem for a non-degenerate orbit Γ( q ) of critical points of U. More precisely, with the additional assumption that the isotropygroup Γ q = { γ ∈ Γ : γq = q } is trivial and that there is at least one positive eigenvalue ofthe Hessian ∇ U ( q ) , we have proved the existence of non-stationary periodic solutions of thesystem (1.1) in any neighborhood of the orbit Γ( q ) . Moreover, we have controlled the minimalperiods of these solutions in terms of positive eigenvalues of ∇ U ( q ) , see Theorem 1.1 of [12].In [13] we have proved the symmetric Liapunov center theorem for a minimal orbit Γ( q ) . We have assumed that Γ( q ) is an isolated orbit of critical points of U which is also an orbit ofminima of U and that the isotropy group Γ q is trivial. Requiring that there is at least one positiveeigenvalue of ∇ U ( q ) we have proved the existence of non-stationary periodic solutions of thesystem (1.1) in any neighborhood of the orbit Γ( q ) . Moreover, we have controlled the minimalperiods of these solutions in terms of positive eigenvalues of ∇ U ( q ) , see Theorem 1.1 of [13].We emphasize that in this theorem the orbit Γ( q ) can be degenerate, i.e. dim ker ∇ U ( q ) > dim Γ( q ) . Since the orbit Γ( q ) is Γ -homeomorphic to Γ / Γ q and in both theorems discussed above wehave assumed that the isotropy group Γ q is trivial, the orbit Γ( q ) is Γ -homeomorphic to thegroup Γ . As far as we know, there is no symmetric Liapunov center theorem for an orbit Γ( q ) ofdimension at least with nontrivial isotropy group Γ q . Therefore the aim of this article is toprove two versions of such theorems.Let l ∈ N ∪ { } and by T l we understand the l -dimensional torus, i.e. T l = { e } , if l = 0 S × · · · × S | {z } l -times , if l = 0 . Let H and K be arbitrary groups. We write H ≈ K if the group H is isomorphic to the group K .Note that in both theorems formulated below if the group Γ is abelian then the isotropygroup Γ q can be arbitrary. On the other hand, if the group of symmetries Γ is not abelian weassume that the isotropy group Γ q is isomorphic to a torus. A natural question arises whetherthese theorems can be strengthened by assuming that the isotropy group of q is arbitrary. Thisquestion is at present far from being solved.The following theorems significantly extend the class of potential applications. For example,if Γ = SO (3) and the isotropy group Γ q is isomorphic to the circle group SO (2) ≈ S , then thefollowing spaces are homeomorphic: Γ( q ) , Γ / Γ q , SO (3) /SO (2) and S , i.e. the orbit Γ( q ) ishomeomorphic to the two-dimensional sphere S . We underline that this case is not covered bytheorems proved in [12, 13].The theorem below is an extension of Theorem 1.1 of [12], which we obtain assuming that l = 0 , i.e. Γ q = { e } . Theorem 1.1. [Symmetric Liapunov center theorem for a non-degenerate orbit] Let Ω ⊂ R n be an open and Γ -invariant subset of an orthogonal representation R n of a compact Lie group ERIODIC ORBITS 3 Γ . Assume that U : Ω → R is a Γ -invariant potential of class C and q ∈ Ω ∩ ( ∇ U ) − (0) . Ifmoreover, (1) Γ is abelian or Γ q ≈ T l for some l ∈ N ∪ { } , (2) dim ker ∇ U ( q ) = dim Γ( q ) , (3) σ ( ∇ U ( q )) ∩ (0 , + ∞ ) = { β , . . . , β m } , β > β > . . . > β m > and m ≥ , then for any β j such that β j /β j N for j = j , there exists a sequence ( q k ( t )) of periodicsolutions of the system (1.1) with a sequence ( T k ) of minimal periods such that T k → π/β j andfor any ε > there exists k ∈ N such that q k ([0 , T k ]) ⊂ Γ( q ) ε for all k ≥ k . The following theorem is a generalization of Theorem 1.1 of [13], which we obtain putting l = 0 , i.e. Γ q = { e } . Theorem 1.2. [Symmetric Liapunov center theorem for a minimal orbit] Let Ω ⊂ R n be an openand Γ -invariant subset of an orthogonal representation R n of a compact Lie group Γ . Assumethat U : Ω → R is a Γ -invariant potential of class C and q ∈ Ω ∩ ( ∇ U ) − (0) . If moreover, (1) Γ is abelian or Γ q ≈ T l for some l ∈ N ∪ { } , (2) Γ( q ) consists of minima of the potential U , (3) Γ( q ) is isolated in ( ∇ U ) − (0) , (4) σ ( ∇ U ( q )) ∩ (0 , + ∞ ) = { β , . . . , β m } , β > β > . . . > β m > and m ≥ , then for any β j such that β j /β j N for j = j , there exists a sequence ( q k ( t )) of periodicsolutions of the system (1.1) with a sequence ( T k ) of minimal periods such that T k → π/β j andfor any ε > there exists k ∈ N such that q k ([0 , T k ]) ⊂ Γ( q ) ε for all k ≥ k . How do we prove these theorems? As in articles [12, 13], we consider periodic solutions ofthe system (1.1) as orbits of critical points of a (Γ × S ) -invariant functional defined on suitablechosen infinite-dimensional orthogonal representation H π of the group Γ × S . To prove ourtheorems we apply techniques of equivariant bifurcation theory. To be more precise, we provea change of the (Γ × S ) -equivariant Conley index, see [8], along the family of trivial orbits Γ( q ) × (0 , + ∞ ) ⊂ H π × (0 , + ∞ ) which implies bifurcation of non-stationary periodic solutionsof the system (1.1).Suppose that Γ q ≈ T l for some l ∈ N . Since the pair (Γ × S , Γ q × S ) is not admissible,the homomorphism i ⋆ : U (Γ q × S ) → U (Γ × S ) of the Euler rings induced by the inclusionhomomorphism i : Γ q × S → Γ × S is not an injection. Therefore, proving the theoremsformulated above, we must perform more subtle and advanced calculations in the Euler ring U (Γ q × S ) than those which were done in articles [12, 13].After introduction our paper is organized as follows. In Section 2 we introduce the equivariantsetting and review some of standard facts on equivariant topology and representation theory ofcompact Lie groups. In Subsection 2.1 we recall the definition of the Euler ring U ( G ) of a compactLie group G. Moreover, the properties of the equivariant Euler characteristic χ G ( X ) ∈ U ( G ) ofa finite pointed G - CW -complex X are also discussed in this subsection. In Subsection 2.2 we lookmore closely at the H -equivariant Euler characteristic χ H ( S V ) ∈ U ( H ) of the H - CW -complex S V where V is an orthogonal representation of H ≈ T l . Section 3 contains the proofs of ourmain results. Subsection 3.1 is dedicated to introducing the variational setting of the problem.Namely, we study periodic solutions of the system (1.1) as orbits of critical points of a (Γ × S ) -invariant functional defined on the Hilbert space H π . Sections 3.2 and 3.3 are devoted to theproofs of Theorems 1.1 and 1.2, respectively.
MARTA KOWALCZYK , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI preliminary results Groups and their representations.
In this section for the convenience of the readerwe repeat the relevant material from [9] and [16] without proofs, thus making our expositionself-contained.Let G stand for a compact Lie group and V = ( R n , ς ) be a finite-dimensional, real, orthogonalrepresentation of G , that is a pair consisting of the space R n and a continuous homomorphism ς : G → O ( n ) , where O ( n ) denotes the group of orthogonal matrices. We call V trivial if ς ( g ) = Id n for any g ∈ G, where Id n is the identity matrix. The linear action of G on V is givenby G × V ∋ ( g, v ) ς ( g ) v ∈ V . To shorten notation, we continue to write gv for ς ( g ) v and by v ∈ V we understand v ∈ R n . A subset Ω ⊂ V is called G -invariant if for any g ∈ G and v ∈ Ω wehave gv ∈ Ω . By an orthogonal subrepresentation of V we understand a linear subspace W ⊂ V which is also a G -invariant set. Additionally, define V G = { v ∈ V : gv = v ∀ g ∈ G } . Two orthogonal representations of G, V = ( R n , ς ) and V ′ = ( R n , ς ′ ) are equivalent if thereexists a G -equivariant, linear isomorphism L : V → V ′ , i.e. the isomorphism L satisfying L ( gv ) = gL ( v ) for any g ∈ G and v ∈ V . We denote it briefly by V ≈ G V ′ . Let ( · , · ) and k·k denote the standard scalar product and the standard norm on R n , respectively.For an orthogonal subrepresentation W ⊂ V we define the orthogonal complement W ⊥ of W as W ⊥ = { v ∈ V : ( v, w ) = 0 ∀ w ∈ W } ⊂ V . By the sum of two orthogonal representations of G, V = ( R n , ς ) and V = ( R n , ς ) weunderstand the representation V ⊕ V , i.e. ( R n + n , ς ⊕ ς ) where the continuous homomorphism ς ⊕ ς : G → O ( n + n ) is given by ( ς ⊕ ς )( g ) = diag ( ς ( g ) , ς ( g )) , g ∈ G. Fix v ∈ V G and define B ε ( V , v ) = { v ∈ V : k v − v k < ε } , D ε ( V , v ) = cl B ε ( V , v ) ,S ε ( V , v ) = ∂D ε ( V , v ) and S V ε,v = D ε ( V , v ) /S ε ( V , v ) . Since V is an orthogonal representationof G and v ∈ V G , the sets B ε ( V , v ) , D ε ( V , v ) , S ε ( V , v ) and S V ε,v are G -invariant. For sim-plicity of notation, we write B ε ( V ) , D ε ( V ) , S ε ( V ) and S V ε for v = 0 and B ( V ) , D ( V ) , S ( V ) and S V for v = 0 and ε = 1 . Let sub( G ) denote the set of closed subgroups of G. Two subgroups
H, K ∈ sub( G ) are calledconjugate in G if there exists g ∈ G such that H = gKg − . Conjugacy is an equivalence relationand the conjugacy class of H ∈ sub( G ) is denoted by ( H ) G . Moreover, sub[ G ] denotes the set ofthe conjugacy classes of closed subgroups of G. If v ∈ V then G v = { g ∈ G : gv = v } ∈ sub( G ) is the isotropy group of v and G ( v ) = { gv : g ∈ G } ⊂ V is the G -orbit through v. Notice that the isotropy groups of points on the same G -orbitare conjugate in G. Fix k, l ∈ N ∪ {∞} and an open, G -invariant subset Ω ⊂ V . A map ϕ : Ω → R of class C k is said to be a G -invariant C k -potential if ϕ ( gv ) = ϕ ( v ) for any g ∈ G and v ∈ Ω . The setof G -invariant C k -potentials is denoted by C kG (Ω , R ) . A map ψ : Ω → V of class C l is calleda G -equivariant C l -map if ψ ( gv ) = gψ ( v ) for any g ∈ G and v ∈ Ω . The set of G -equivariant C l -maps is denoted by C lG (Ω , V ) . For any G -invariant C k -potential ϕ the gradient of ϕ denotedby ∇ ϕ is G -equivariant C k − -map. Similarly, we use the symbol ∇ ϕ to denote the Hessian of ϕ. Let us recall the notion of an admissible pair which was introduced in [12].
Definition 2.1.1.
Let H ∈ sub( G ) . A pair ( G, H ) is said to be admissible if for all K , K ∈ sub( H ) the following implication holds true: if ( K ) H = ( K ) H then ( K ) G = ( K ) G . Remark 2.1.1.
Note that a pair ( G, H ) is admissible if for all K , K ∈ sub( H ) the followingequivalence holds: ( K ) H = ( K ) H iff ( K ) G = ( K ) G . ERIODIC ORBITS 5
Remark 2.1.2.
Let G be abelian and H ∈ sub( G ) . Then the pair ( G, H ) is admissible. Indeed,note that for all K , K ∈ sub( H ) we have ( K ) H = ( K ) H iff K = K iff ( K ) G = ( K ) G . Generally speaking, the class of admissible pairs is very restrictive.
Example 2.1.1. If Γ is an abelian compact Lie group, Γ ′ ∈ sub(Γ) , G = Γ × S and H = Γ ′ × S then the pair ( G, H ) is admissible.Let F ∗ ( G ) denote the set of finite, pointed G -CW-complexes, see [16] for the definition of G -CW-complex. The G -homotopy type of X ∈ F ∗ ( G ) is denoted by [ X ] G and F ∗ [ G ] is theset of G -homotopy types of finite, pointed G -CW-complexes. Let F be a free abelian groupgenerated by F ∗ [ G ] and N be a subgroup of F generated by elements [ A ] G − [ X ] G + [ X/A ] G where A, X ∈ F ∗ ( G ) and A ⊂ X. Define U ( G ) = F/N and let χ G ( X ) be the class of an element [ X ] G ∈ F in U ( G ) . If X is a finite G -CW-complex without base point we put χ G ( X ) = χ G ( X + ) where X + = X ⊔ {∗} and ∗ is a separate point added such that g ∗ = ∗ for all g ∈ G. For ( X, ∗ X ) , ( Y, ∗ Y ) ∈ F ∗ ( G ) put X ∨ Y = ( X ×{∗ Y }∪{∗ X }× Y ) / { ( ∗ X , ∗ Y ) } and X ∧ Y = X × Y /X ∨ Y. Then X ∨ Y, X ∧ Y ∈ F ∗ ( G ) . Since [ X ] G − [ X ∨ Y ] G +[ Y ] G = [ X ] G − [ X ∨ Y ] G +[ X ∨ Y /X ] G ∈ N, we have χ G ( X ) + χ G ( Y ) = χ G ( X ∨ Y ) . Additionally, the assignment ( X, Y ) X ∧ Y inducesa product U ( G ) × U ( G ) → U ( G ) given by the formula χ G ( X ) ⋆ χ G ( Y ) = χ G ( X ∧ Y ) . The proof of the following theorem one can find in [16].
Theorem 2.1.1.
The group ( U ( G ) , +) is the free abelian group with basis χ G ( G/H + ) for ( H ) G ∈ sub[ G ] . Moreover, if X ∈ F ∗ ( G ) and p S k =0 { ( k, ( H j,k ) G ) : j = 1 , . . . , q ( k ) } is a type of the celldecomposition of X then χ G ( X ) = P ( H ) G ∈ sub[ G ] η G ( H ) G ( X ) · χ G ( G/H + ) ∈ U ( G ) where η G ( H ) G ( X ) = p P k =0 ( − k ν ( X, k, ( H ) G ) ∈ Z and ν ( X, k, ( H ) G ) is the number of cells of dimension k and of orbittype ( H ) G in X. The triple ( U ( G ) , + , ⋆ ) is a commutative ring with unity I U ( G ) = χ G ( G/G + ) and it is calledthe Euler ring of G, see [15, 16] for more properties of U ( G ) . Let H ∈ sub( G ) and Y be a H -space, see [16] for the definition of H -space. Now define anaction of H on the product G × Y by the formula ( h, ( g, y )) ( gh − , hy ) and let G × H Y denotethe space of H -orbits of this action. We denote the H -orbit through ( g, y ) briefly by [ g, y ] . Thespace G × H Y is a G -space with the following action ( g ′ , [ g, y ]) [ g ′ g, y ] . For a pointed H -space Y and G + = G ⊔ {∗} where ∗ is a separate point added such that g ∗ = ∗ for all g ∈ G we have G + ∧ Y = G + × Y /G + ∨ Y = G × Y /G × {∗} . The space G + ∧ Y is a pointed H -space withthe following action ( h, ( g, y )) ( gh − , hy ) and by G + ∧ H Y we denote the orbit space of thisaction. Similarly, we write the H -orbit through ( g, y ) as [ g, y ] . Note that G + ∧ H Y is a pointed G -space with an action induced by the assignment ( g ′ , [ g, y ]) [ g ′ g, y ] . The point of the following theorem is that it allows to express the G -equivariant Euler char-acteristic of a G -CW-complex G + ∧ H Y in terms of the H -equivariant Euler characteristic of the H -CW-complex Y. The theorem below was proved in [12], see Theorems 2.2 and 2.3 of [12].
Theorem 2.1.2.
Fix H ∈ sub( G ) and Y ∈ F ∗ ( H ) . If χ H ( Y ) = P ( K ) H ∈ sub [ H ] η H ( K ) H ( Y ) · χ H ( H/K + ) then (1) G + ∧ H Y ∈ F ∗ ( G ) , MARTA KOWALCZYK , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI (2) χ G ( G + ∧ H Y ) = X ( K ) H ∈ sub[ H ] η H ( K ) H ( Y ) · χ G ( G/K + ) = X ( K ) G ∈ sub[ G ] η G ( K ) G ( G + ∧ H Y ) · χ G ( G/K + ) ∈ U ( G ) and η G ( K ) G ( G + ∧ H Y ) = P ( K ) H ∈ sub[ H ] , ( K ) G =( K ) G η H ( K ) H ( Y ) ∈ Z , (3) if moreover, the pair ( G, H ) is admissible then η G ( K ) G ( G + ∧ H Y ) = η H ( K ) H ( Y ) and that iswhy the map U ( H ) ∋ χ H ( Y ) → χ G ( G + ∧ H Y ) ∈ U ( G ) is injective. Corollary 2.1.1. If G is abelian and H ∈ sub( G ) then by Remark 2.1.2 the pair ( G, H ) isadmissible and the map U ( H ) ∋ χ H ( Y ) → χ G ( G + ∧ H Y ) ∈ U ( G ) is injective. Fix an open, G -invariant subset Ω ⊂ V . Let ϕ ∈ C G (Ω , R ) and G ( q ′ ) , G ( q ′′ ) ⊂ Ω be non-degenerate critical G -orbits of ϕ, i.e. G ( q ν ) ⊂ ( ∇ ϕ ) − (0) ∩ Ω and dim ker ∇ ϕ ( q ν ) = dim G ( q ν ) where ν ∈ { ′ , ′′ } . Additionally, assume that G q ′ = G q ′′ = H ∈ sub( G ) . Fix ν ∈ { ′ , ′′ } and note that T q ν V = T q ν G ( q ν ) ⊕ T q ν G ( q ν ) ⊥ where T q ν G ( q ν ) ⊥ is an orthogonalrepresentation of H, and define φ ν = ϕ | T qν G ( q ν ) ⊥ ∈ C H ( T q ν G ( q ν ) ⊥ , R ) . In the following result we express the G -equivariant Conley index CI G ( G ( q ν ) , −∇ ϕ ) of the non-degenerate critical G -orbit G ( q ν ) in terms of the H -equivariant Conley index CI H ( { q ν } , −∇ φ ν ) of the non-degenerate critical point q ν of the potential ϕ restricted to the space orthogonal to thisorbit. So this theorem gains in interest if we realize that this relation allows us to distinguish the G -equivariant Conley indexes of non-degenerate orbits considering only the potential restrictedto the spaces orthogonal to these orbits. The proof of the following theorem one can find in [12]. Theorem 2.1.3.
Under the above assumptions, if ν ∈ { ′ , ′′ } then (1) CI H ( { q ν } , −∇ φ ν ) ∈ F ∗ [ H ] , (2) CI G ( G ( q ν ) , −∇ ϕ ) = G + ∧ H CI H ( { q ν } , −∇ φ ν ) ∈ F ∗ [ G ] , (3) if the pair ( G, H ) is admissible and if χ H ( CI H ( { q ′ } , −∇ φ ′ )) = χ H ( CI H ( { q ′′ } , −∇ φ ′′ )) ∈ U ( H ) , then CI G ( G ( q ′ ) , −∇ ϕ ) = CI G ( G ( q ′′ ) , −∇ ϕ ) ∈ F ∗ [ G ] and χ G ( CI G ( G ( q ′ ) , −∇ ϕ )) = χ G ( CI G ( G ( q ′′ ) , −∇ ϕ )) ∈ U ( G ) . Remark 2.1.3.
Let H ∈ sub( G ) . The standard homomorphism i : H → G induces the ringhomomorphism i ⋆ : U ( G ) → U ( H ) . Rabinowitz proved that the Brouwer index of an isolated critical point which is a local min-imum of a potential of class C equals ∈ Z , see Lemma 1.1 of [14]. The following lemma isan analogue of the Rabinowitz result for the class of equivariant gradient maps. Instead of theBrouwer degree we use the degree for equivariant gradient maps ∇ H - deg( · , · ) ∈ U ( H ) , see [4]for the definition and properties of this degree. In the proof of the lemma below we use therelation between the degree for equivariant gradient maps and the equivariant Conley index, see[4], instead of the Poincaré-Hopf theorem used by Rabinowitz. Lemma 2.1.1.
Let V be an orthogonal representation of a compact Lie group H and ϕ ∈ C H ( V , R ) . Assume that ∈ V is an isolated critical point and local minimum of ϕ. Then ∇ H - deg( ∇ ϕ, B ε ( V )) = I U ( H ) ∈ U ( H ) . Proof.
There is no loss of generality in assuming that ϕ (0) = 0 . Since ∈ V is an isolatedcritical point of H -invariant potential ϕ, { } ⊂ V is an isolated invariant set in the sense of the H -equivariant Conley index theory. Therefore the H -equivariant Conley index CI H ( { } , −∇ ϕ ) is well defined. Since ∈ V is an isolated critical point of ϕ , we can fix ε > such that ∇ ϕ ( v ) = 0 for D ε ( V ) \{ } . Choose < c < min v ∈ S ε ( V ) ϕ ( v ) a regular value of ϕ and define A c = ERIODIC ORBITS 7 ϕ − (( −∞ , c ]) ∩ D ǫ ( V ) . Since ϕ (0) = 0 , ∈ A c . Moreover, by the choice of the regular value c we have A c ⊂ B ε ( V ) and ∂A c = ϕ − ( c ) ∩ D ε ( V ) is a manifold of codimension . Since ∈ V is an isolated minimum of ϕ , A c is contractible to a point by using the negative gradient flowcorresponding to ϕ. Therefore the pair ( A c , ∅ ) is a H -index pair for the isolated invariant set { } . Summing up, we obtain χ H ( CI H ( { } , −∇ ϕ )) = ( − χ H ( H/H + ) = I U ( H ) ∈ U ( H ) . Applyingthe equality χ H ( CI H ( { } , −∇ ϕ )) = ∇ H - deg( ∇ ϕ, B ε ( V )) , see [4], we complete the proof. (cid:3) The following lemma is analogous to Lemma 2.1.1. But instead of minimum of the potential ϕ we consider its maximum. Lemma 2.1.2.
Let V be an orthogonal representation of H and ϕ ∈ C H ( V , R ) . Assume that ∈ V is an isolated critical point and local maximum of ϕ. Then ∇ H - deg( ∇ ϕ, B ε ( V )) = χ H ( S V ) ∈ U ( H ) . Proof.
Without loss of generality we can assume that ϕ (0) = 0 . Like in the proof of Lemma2.1.1 it follows that the H -equivariant Conley index CI H ( { } , −∇ ϕ ) is well defined. Since ∈ V is an isolated critical point of ϕ, we can choose ε > such that ∇ ϕ ( v ) = 0 for D ε ( V ) \{ } . Choose max v ∈ S ε ( V ) ϕ ( v ) < − c < a regular value of ϕ and define A c = ϕ − ([ − c, + ∞ )) ∩ D ε ( V ) . By the choice of the regular value − c we obtain that A c ⊂ B ε ( V ) and ∈ A c . Since − c is theregular value of ϕ, we have ∂A c = ϕ − ( − c ) ∩ D ε ( V ) is a manifold of codimension . Moreover,since ∈ V is a maximum of ϕ , the negative gradient flow corresponding to ϕ is directedoutwards on ∂A c . Additionally, A c is contractible by using the gradient flow correspondingto ϕ. Hence the pair ( A c , ∂A c ) is a H -index pair for the isolated invariant set { } . It followsthat χ H ( CI H ( { } , −∇ ϕ )) = χ H ( A c /∂A c ) = χ H ( D ( V ) /S ( V )) = χ H ( S V ) ∈ U ( H ) . Applying theequality χ H ( CI H ( { } , −∇ ϕ )) = ∇ H - deg( ∇ ϕ, B ε ( V )) , see [4], we complete the proof. (cid:3) Torus.
Let l ∈ N ∪ { } and recall that T l = { e } , if l = 0 S × · · · × S | {z } l -times , if l = 0 . By e iφ ∈ T l we mean e iφ = ( e iφ , . . . , e iφ l ) for some φ = ( φ , . . . , φ l ) , φ i ∈ [0 , π ) , i = 1 , . . . , l. Fix m ∈ Z l and define H m = { e iφ ∈ T l : e i ( m,φ ) = 1 } = { e iφ ∈ T l : ( m, φ ) ∈ π Z } ∈ sub( T l ) . Since T l is an abelian group, ( H m ) T l = H m . Notice that H m = H m ′ if and only if m = ± m ′ for every m, m ′ ∈ Z l . Fix m ∈ Z l \{ } and let homomorphisms ς m : T l → O (2) and ς : T l → O (1) be given by ς m ( e iφ ) = (cid:20) cos( m, φ ) − sin( m, φ )sin( m, φ ) cos( m, φ ) (cid:21) , ς ( e iφ ) = 1 . (2.2.1)Write ς km = diag ( ς m , . . . , ς m | {z } k -times ) and ς k = diag ( ς , . . . , ς | {z } k -times ) for some k ∈ N . The following theorem gives the classification of irreducible representations of the torus T l , see [10]. Theorem 2.2.1.
The real, irreducible representations of the torus T l are the following: (1) R [1 , m ] = ( R , ς m ) , m ∈ Z l \{ } , (2) R [1 ,
0] = ( R , ς ) . MARTA KOWALCZYK , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI Representations R [1 , m ] and R [1 , m ′ ] are equivalent if and only if m = ± m ′ . Moreover, T lv = (cid:26) H m , if v ∈ R [1 , m ] \{ } T l , if v = 0 . Let G be a compact Lie group and H ∈ sub( G ) . From now on we assume that there is l > such that H ≈ T l and that V is an orthogonalrepresentation of H . Then there exist r, k ∈ N ∪ { } , k , . . . , k r ∈ N and m , . . . , m r ∈ Z l \{ } such that V ≈ H ( R n , ς ) = R [ k , ⊕ R [ k , m ] ⊕ · · · ⊕ R [ k r , m r ] where m i = ± m j for i = j, i.e. ς = diag ( ς k , ς k m , . . . , ς k r m r ) . The theorem below yields partial information about the H -equivariant Euler characteristic ofthe H -CW-complex S V . The proof of this theorem one can find in [10].
Theorem 2.2.2.
Under the above assumptions, the following equality holds χ H ( S V ) = ( − k χ H ( H/H + ) − r X i =1 k i · χ H ( H/H + m i ) ! ++ X ( K ) H ∈{ ( H ) H ∈ sub[ H ]:dim H≤ l − } η H ( K ) H ( S V ) · χ H ( H/K + ) ∈ U ( H ) . (2.2.2)Now we define a number S ( S V ) as the sum of absolute values of the coefficients of χ H ( S V ) assigned to generators χ H ( H/K + ) of U ( H ) such that dim K = l − . Note that the onlygenerators which satisfy this condition and have non-zero coefficients in χ H ( S V ) are of the form χ H ( H/H + m i ) for i = 1 , . . . , r. Definition 2.2.1.
Define S ( S V ) by S ( S V ) = r P i =1 k i ∈ N . It is understood that if V is a trivialrepresentation of H we put S ( S V ) = 0 . Let W be an orthogonal representation of H equivalent to R [ k ′ , ⊕ R [ k ′ , m ′ ] ⊕ · · · ⊕ R [ k ′ s , m ′ s ] for some s, k ′ ∈ N ∪ { } , k ′ , . . . , k ′ s ∈ N and m ′ , . . . , m ′ s ∈ Z l \{ } where m ′ i = ± m ′ j for i = j. Remark 2.2.1.
The decompositions of the orthogonal representations of H, V and W are uniqueup to order of elements. Note that V ⊕ W is equivalent to R [ k + k ′ , ⊕ R [ k , m ] ⊕ · · · ⊕ R [ k r , m r ] ⊕ R [ k ′ , m ′ ] ⊕ · · · ⊕ R [ k ′ s , m ′ s ] . It is clear that there exist t, k ′′ ∈ N ∪ { } , k ′′ , . . . , k ′′ t ∈ N and m ′′ , . . . , m ′′ t ∈ Z l \{ } such that V ⊕ W ≈ H R [ k ′′ , ⊕ R [ k ′′ , m ′′ ] ⊕· · ·⊕ R [ k ′′ t , m ′′ t ] , k ′′ = k + k ′ and t P i =1 k ′′ i = r P i =1 k i + s P i =1 k ′ i where m ′′ i = ± m ′′ j for i = j. Lemma 2.2.1.
Under the above assumptions, the following conditions hold. (1)
The number S ( S W ) = 0 if and only if W is a trivial representation of H, i.e. W ≈ H R [ k ′ , . (2) S ( S V ⊕ W ) = S ( S V ) + S ( S W ) . (3) If W is a nontrivial representation of H then S ( S V ) = S ( S V ⊕ W ) . ERIODIC ORBITS 9
Proof.
Condition (1) is obvious and condition (3) follows from (1) and (2). We only need toshow (2). First of all, by Theorem 2.2.2 and Remark 2.2.1, we have χ H ( S V ⊕ W ) = ( − k ′′ χ H ( H/H + ) − t X i =1 k ′′ i · χ H ( H/H + m ′′ i ) ! ++ X ( K ) H ∈{ ( H ) H ∈ sub[ H ]:dim H≤ l − } η H ( K ) H ( S V ⊕ W ) · χ H ( H/K + ) and t P i =1 k ′′ i = r P i =1 k i + s P i =1 k ′ i . Then S ( S V ⊕ W ) = t P i =1 k ′′ i = S ( S V ) + S ( S W ) , which is our claim. (cid:3) The following lemma can be easily deduced from Corollary 3.2 of [10].
Lemma 2.2.2.
The following conditions are equivalent: (1) χ H ( S V ) = χ H ( S V ⊕ W ) , (2) W is a trivial and even-dimensional representation of H . We can now formulate the relation between the number S ( S V ) and the H -equivariant Eulercharacteristic χ H ( S V ) . Lemma 2.2.3. If S ( S V ) = S ( S W ) then χ H ( S V ) = χ H ( S W ) . Proof.
Without loss of generality we assume that S ( S V ) = 0 . Since S ( S V ) = 0 , V is a nontrivialrepresentation of H. Suppose, contrary to our claim, that S ( S V ) = S ( S W ) and χ H ( S V ) = χ H ( S W ) . Since the representation V is nontrivial and χ H ( S V ) = χ H ( S W ) , it follows that W isa nontrivial representation of H. By Theorem 2.2.2, we have χ H ( S V ) =( − k χ H ( H/H + ) − r X i =1 k i · χ H ( H/H + m i ) ! ++ X ( K ) H ∈{ ( H ) H ∈ sub[ H ]:dim H≤ l − } η H ( K ) H ( S V ) · χ H ( H/K + ) ,χ H ( S W ) =( − k ′ χ H ( H/H + ) − s X i =1 k ′ i · χ H ( H/H + m ′ i ) ! ++ X ( K ) H ∈{ ( H ) H ∈ sub[ H ]:dim H≤ l − } η H ( K ) H ( S W ) · χ H ( H/K + ) . Therefore the equality χ H ( S V ) = χ H ( S W ) implies that k and k ′ are of the same parity and r X i =1 k i · χ H ( H/H + m i ) = s X i =1 k ′ i · χ H ( H/H + m ′ i ) . (2.2.3)Note that the sets { χ H ( H/H + m ′ ) , . . . , χ H ( H/H + m ′ s ) } and { χ H ( H/H + m ) , . . . , χ H ( H/H + m r ) } consistof linearly independent elements of U ( H ) . We claim that { χ H ( H/H + m ′ ) , . . . , χ H ( H/H + m ′ s ) } = { χ H ( H/H + m ) , . . . , χ H ( H/H + m r ) } , r = s and { k ′ , . . . , k ′ s } = { k , . . . , k r } . Suppose, contrary to ourclaim, that there exists χ H ( H/H + m ′ i )
6∈ { χ H ( H/H + m ) , . . . , χ H ( H/H + m r ) } . Hence χ H ( H/H + m ′ i ) / ∈ s [ i =1 ,i = i { χ H ( H/H + m ′ i ) } ∪ r [ i =1 { χ H ( H/H + m i ) } . By the equality (2.2.3), we obtain k ′ i χ H ( H/H + m ′ i ) = , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI r P i =1 k i · χ H ( H/H + m i ) − s P i =1 ,i = i k ′ i · χ H ( H/H + m ′ i ) , i.e. the basis element χ H ( H/H + m ′ i ) is presented asa linear combination of basis elements, see Theorem 2.1.1, a contradiction. We have just provedthat { χ H ( H/H + m ′ ) , . . . , χ H ( H/H + m ′ s ) } ⊂ { χ H ( H/H + m ) , . . . , χ H ( H/H + m r ) } . The reverse conclusion is proved in the same way. Since { χ H ( H/H + m ′ ) , . . . , χ H ( H/H + m ′ s ) } = { χ H ( H/H + m ) , . . . , χ H ( H/H + m r ) } and by the equality (2.2.3), we have r = s, for all j = 1 , . . . , r,k j = k ′ j and m j = ± m ′ j . In consequence, S ( S V ) = r P i =1 k i = S ( S W ) , a contradiction. (cid:3) As a direct consequence of Lemmas 2.2.1.(3) and 2.2.3 we obtain the following corollary.
Corollary 2.2.1. If W is a nontrivial representation of H then χ H ( S V ) = χ H ( S V ⊕ W ) . The crucial role in the following theorem plays the connection of the G -equivariant Eulercharacteristic of the G -CW-complex G + ∧ H S V with the H -equivariant Euler characteristic ofthe H -CW-complex S V , see Theorem 2.1.2. Notice that the pair ( G, H ) does not need to beadmissible, i.e. it can happen that there exist two subgroups K , K ∈ sub( G ) such that ( K ) G =( K ) G and ( K ) H = ( K ) H . Theorem 2.2.3. If W is a nontrivial representation of H then the following inequality holdstrue χ G ( G + ∧ H S V ) = χ G ( G + ∧ H S V ⊕ W ) . Proof.
On the contrary, suppose that W is a nontrivial representation of H and χ G ( G + ∧ H S V ) = χ G ( G + ∧ H S V ⊕ W ) . By Lemma 2.2.1.(3), we get S ( S V ) = S ( S V ⊕ W ) . Note that the operation ofconjunction preserves the dimensions of subgroups, i.e. dim K = dim gKg − for every g ∈ G and K ∈ sub( G ) . Thus Theorem 2.1.2 now leads to χ G ( G + ∧ H S V ) =( − k χ G ( G/H + ) − r X i =1 k i · χ G ( G/H + m i ) ! ++ X ( K ) G ∈{ ( H ) G ∈ sub[ G ]:dim H≤ l − } η G ( K ) G ( G + ∧ H S V ) · χ G ( G/K + ) ,χ G ( G + ∧ H S V ⊕ W ) =( − k + k ′ χ G ( G/H + ) − r X i =1 k i · χ G ( G/H + m i ) − s X j =1 k ′ j · χ G ( G/H + m ′ j ) ++ X ( K ) G ∈{ ( H ) G ∈ sub[ G ]:dim H≤ l − } η G ( K ) G ( G + ∧ H S V ⊕ W ) · χ G ( G/K + ) , and so k ′ is an even number. The equality χ G ( G + ∧ H S V ) = χ G ( G + ∧ H S V ⊕ W ) implies that r X i =1 k i · χ G ( G/H + m i ) = r X i =1 k i · χ G ( G/H + m i ) + s X j =1 k ′ j · χ G ( G/H + m ′ j ) , ERIODIC ORBITS 11 and consequently s P j =1 k ′ j · χ G ( G/H + m ′ j ) = 0 ∈ U ( G ) . By Theorem 2.1.1, we obtain k ′ j = 0 for any j, which gives S ( S V ) = r P i =1 k i = S ( S V ⊕ W ) , a contradiction. (cid:3) Recall that V is an orthogonal representation of T l . Define V k = { a cos kt + b sin kt : a, b ∈ V } , k ≥ . The action of the group T l × S on V k is defined as follows ( T l × S ) × V k ∋ (( e iφ , e iθ ) , a cos kt + b sin kt ) ς ( e iφ ) a cos k ( t + θ ) + ς ( e iφ ) b sin k ( t + θ ) . Thus V k is an orthogonal representation of T l × S . Lemma 2.2.4.
Under the above assumptions, V T l × S k = { } , k ≥ . Proof.
Notice that ∈ V T l × S k . Let a cos kt + b sin kt ∈ V T l × S k , i.e. for all ( e iφ , e iθ ) ∈ T l × S we have ( e iφ , e iθ )( a cos kt + b sin kt ) = a cos kt + b sin kt. Therefore ς ( e iφ ) a cos kθ + ς ( e iφ ) b sin kθ = a and − ς ( e iφ ) a sin kθ + ς ( e iφ ) b cos kθ = b. (2.2.4)Since the equalities (2.2.4) are satisfied for all φ and θ, for φ = 0 and θ = π/k we get a = b = 0 , and the proof is complete. (cid:3) Proofs of Theorems 1.1 and 1.2
To prove our main results we use techniques of equivariant bifurcation theory. Set G = Γ × S . We treat periodic solutions of the system (1.1) as G -orbits of critical points of a G -invariantpotential. As a topological tool we use the G -equivariant Conley index due to Izydorek, see [8].More precisely, we prove a change of the G -equivariant Conley index along the trivial family Γ( q ) × (0 , + ∞ ) ⊂ H π × (0 , + ∞ ) . Such a change implies a local bifurcation of periodic solutionsof the system ¨ q ( t ) = −∇ U ( q ( t )) . Variational setting.
In this section we introduce the variational setting for our problem,i.e. we consider periodic solutions of the system (1.1) as critical G -orbits of G -invariant func-tionals. It is known that there is a one-to-one correspondence between πλ -periodic solutions ofthe system (1.1) and π -periodic solutions of the following family ¨ q ( t ) = − λ ∇ U ( q ( t )) q (0) = q (2 π )˙ q (0) = ˙ q (2 π ) . (3.1.1)Write H π = { u : [0 , π ] → R n : u is abs. continuous map, u (0) = u (2 π ) , ˙ u ∈ L ([0 , π ] , R n ) } and h u, v i H π = Z π ( ˙ u ( t ) , ˙ v ( t ))+( u ( t ) , v ( t )) dt. Then (cid:16) H π , h· , ·i H π (cid:17) is a separable Hilbert spacewhich is also an orthogonal representation of G with the following action G × H π ∋ (( γ, e iθ ) , q ( t )) (cid:26) γq ( t + θ ) , if t + θ < πγq ( t + θ − π ) , if t + θ ≥ π . By k · k H π we denote the norm induced by the inner product h· , ·i H π . Let
Φ : H π × (0 , + ∞ ) → R be given by the formula Φ( q, λ ) = Z π (cid:18) k ˙ q ( t ) k − λ U ( q ( t )) (cid:19) dt, (3.1.2) , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI where λ is treated as a parameter. Then Φ is a G -invariant functional of class C and solutionsof the system (3.1.1) correspond to solutions of the following equation ∇ q Φ( q, λ ) = 0 . (3.1.3)Write H = R n and H k = { a cos kt + b sin kt : a, b ∈ R n } for k > . Note that H π = H ⊕ ∞ M k =1 H k (3.1.4)where the finite-dimensional space H k is an orthogonal representation of G for k ≥ . Since q ∈ H π is a constant function, T = G ( q ) × (0 , + ∞ ) = Γ( q ) × (0 , + ∞ ) ⊂ H π × (0 , + ∞ ) . The family T is called a family of trivial solutions of the equation (3.1.3) while we call N = { ( q, λ ) ∈ H π × (0 , + ∞ ) \ T : ∇ q Φ( q, λ ) = 0 } a set of nontrivial solutions.Now we look for parameters which satisfy the necessary condition for the existence of localbifurcation, i.e. ker ∇ q Φ( q , λ ) ∩ ∞ M k =1 H k = ∅ , (3.1.5)see Section 4 of [12] and Theorem 3.2.1 of [13]. The condition (3.1.5) is fulfilled if and only if k − λ β j = 0 for some k ∈ N , see Lemma 5.1.1 of [3]. Let Λ = n kβ j : k ∈ N , j = 1 , . . . , m o andthen a local bifurcation of solutions of the equation (3.1.3) from the trivial family T can occuronly from orbits Γ( q ) × Λ ⊂ T . Fix ε > . By G ( q ) ε and Γ( q ) ε we understand ε -neighborhoods of G ( q ) and Γ( q ) in H π and H = R n , respectively, i.e. G ( q ) ε = [ q ∈ G ( q ) B ε ( H π , q ) ⊂ H π and Γ( q ) ε = [ q ∈ Γ( q ) B ε ( H , q ) ⊂ H = R n . It is known that a change of the G -equivariant Conley index CI G ( G ( q ) , −∇ Φ( · , λ )) along thetrivial family T implies the existence of a local bifurcation of solutions of the equation (3.1.3),where CI G ( G ( q ) , −∇ Φ( · , λ )) is an infinite-dimensional generalization of the G -equivariant Con-ley index which is a G -homotopy type of a G -spectrum. This construction is due to Izydorek,see [8] for more details.Notice that the local bifurcations mentioned above are bifurcations in the function space H π but more interesting phenomena are bifurcations in the phase space, and these kinds ofbifurcations are the claims of our Theorems 1.1 and 1.2. The following lemma states that theexistence of a local bifurcation in the phase space is a natural consequence of the occurrence ofa local bifurcation in the function space. Lemma 3.1.1.
Under the above assumptions, if there exists a local bifurcation of solutions ofthe equation (3.1.3) from the critical orbit G ( q ) , i.e. there exists a sequence ( q k ( t )) of periodicsolutions of the equation (1.1) such that for any ε > there exists k ∈ N such that q k ∈ G ( q ) ε for all k ≥ k , then for any ε > there exists k ∈ N such that q k ([0 , T k ]) ⊂ Γ( q ) ε for all k ≥ k . Proof.
Fix ε > . Since there exists k ∈ N such that q k ∈ G ( q ) ε for all k ≥ k and q isa constant function, we obtain γ ∈ Γ such that q k ∈ B ε ( H π , γq ) , that is k q k − γq k H π < ε. Therefore, by Proposition 1.1 of [11], for some c > we have sup t ∈ [0 ,T k ] k q k ( t ) − γq k = k q k − γq k ∞ ≤ c k q k − γq k H π < cε. In consequence q k ([0 , T k ]) ⊂ Γ( q ) cε for all k ≥ k , which completes theproof. (cid:3) ERIODIC ORBITS 13
So to prove Theorems 1.1 and 1.2 we need to show a change in the G -equivariant Conley indexof the G -orbit G ( q ) . Proof of Theorem 1.1.
We follow the notation used in [12]. Fix β j satisfying the as-sumptions of Theorem 1.1 and choose ε > such that [ λ − , λ + ] ∩ Λ = { β j } where λ ± = ± εβ j . To prove this theorem we have to show that CI G ( G ( q ) , −∇ Φ( · , λ − )) = CI G ( G ( q ) , −∇ Φ( · , +)) . The G -equivariant Conley index CI G ( G ( q ) , −∇ Φ( · , λ ± )) is a G -homotopy type of a G -spectrum ( E n, ± ) ∞ n = n where E n, ± = CI G ( G ( q ) , −∇ Φ n ( · , λ ± )) and Φ n ( · , λ ± ) = Φ( · , λ ± ) |⊕ nk =0 H k : n M k =0 H k → R . It follows that for any n ≥ n we have CI G ( G ( q ) , −∇ Φ n ( · , λ ± )) = CI G ( G ( q ) , −∇ Φ n ( · , λ ± )) , (3.2.1)see [8] for more details.For simplicity of notation we set H = G q = Γ q × S and let Ψ n ± = Ψ ±| H n : H n → R , Ψ ± =Φ( · , λ ± ) | H : H → R and H = T ⊥ q G ( q ) , H n = T ⊥ q Γ( q ) ⊕ n M k =1 H k . Analogously like in Lemma 4.1of [12], using the same H -equivariant gradient homotopy, it follows that for any n ≥ n we have CI H ( { q } , −∇ Ψ n ± ) = CI H ( { q } , −∇ Ψ n ± ) (3.2.2)and CI H ( { q } , −∇ Ψ n ± ) = S H +1 , ± ⊕ W + where the spectral decomposition of H n for the isomor-phism −∇ Ψ n ± ( q ) is the following H n = H ⊕ T ⊥ q Γ( q ) ⊕ n M k =2 H k ! = ( H − , ± ⊕ H +1 , ± ) ⊕ ( W − ⊕ W + ) and dim H +1 , − = dim H +1 , + . From now on we consider two cases: Γ q ≈ T l or Γ is abelian.3.2.1. CASE: Γ q ≈ T l . For l = 0 we have proved this theorem in [12]. Let l > . In thiscase H ≈ T l × S . Analogously like in Lemma 4.1 of [12] we have H +1 , + = H +1 , − ⊕ U where U = { a cos t + b sin t : a, b ∈ V ∇ U ( q ) ( β j ) } and V ∇ U ( q ) ( β j ) is the eigenspace of ∇ U ( q ) corresponding to the eigenvalue β j . Note that V ∇ U ( q ) ( β j ) is an orthogonal representation ofthe torus T l . Since U H = { } , see Lemma 2.2.4, U is a nontrivial, fixed point free, orthogonalrepresentation of H . Thus CI H ( { q } , −∇ Ψ n − ) = S H +1 , − ⊕ W + , CI H ( { q } , −∇ Ψ n + ) = S H +1 , − ⊕ U ⊕ W + and by Theorem 2.2.3 we obtain χ G ( G + ∧ H S H +1 , − ⊕ W + ) = χ G ( G + ∧ H S H +1 , − ⊕ U ⊕ W + ) . Hence byTheorem 2.1.3.(2), the equalities (3.2.1) and (3.2.2) for any n ≥ n we obtain χ G ( CI G ( G ( q ) , −∇ Φ n ( · , λ − ))) = χ G ( G + ∧ H CI H ( { q } , −∇ Ψ n − )) = = χ G ( G + ∧ H CI H ( { q } , −∇ Ψ n + )) = χ G ( CI G ( G ( q ) , −∇ Φ n ( · , λ + ))) , and consequently CI G ( G ( q ) , −∇ Φ n ( · , λ − )) = CI G ( G ( q ) , −∇ Φ n ( · , λ + )) for any n ≥ n , whichcompletes the proof of the first case. , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI CASE: Γ is abelian. By Example 2.1.1, the pair ( G, H ) = (Γ × S , Γ q × S ) is admissible.In view of Theorem 2.1.3.(3) to prove this case it is enough to show that for any n ≥ n χ H ( CI H ( { q } , −∇ Ψ n − )) = χ H ( CI H ( { q } , −∇ Ψ n + )) . Let i ⋆ : U ( H ) → U ( S ) be the ring homomorphism induced by the inclusion i : { e } × S → H, see Remark 2.1.3. Like in the proof of the previous case CI H ( { q } , −∇ Ψ n − ) = S H +1 , − ⊕ W + and CI H ( { q } , −∇ Ψ n + ) = S H +1 , − ⊕ U ⊕ W + where U is a nontrivial representation of S ≈ { e } × S ⊂ H and the action of S is given by shift in time. Since U is a nontrivial representation of S andby the equality (3.2.2), for any n ≥ n we obtain i ⋆ ( χ H ( CI H ( { q } , −∇ Ψ n − ))) = i ⋆ ( χ H ( S H +1 , − ⊕ W + )) = χ S ( S H +1 , − ⊕ W + ) = = χ S ( S H +1 , − ⊕ U ⊕ W + ) = i ⋆ ( χ H ( S H +1 , − ⊕ U ⊕ W + )) = i ⋆ ( χ H ( CI H ( { q } , −∇ Ψ n + ))) , see Lemma 4.1 of [12], and consequently χ H ( CI H ( { q } , −∇ Ψ n − )) = χ H ( CI H ( { q } , −∇ Ψ n + )) forany n ≥ n , which completes the proof of the second case.3.3. Proof of Theorem 1.2.
We follow the notation used in [13]. Fix β j satisfying the as-sumptions of Theorem 1.2 and choose ε > such that [ λ − , λ + ] ∩ Λ = { β j } where λ ± = ± εβ j . To prove this theorem we have to show a change of the G -equivariant Conley index of the orbit G ( q ) at level β j , i.e. CI G ( G ( q ) , −∇ Φ( · , λ − )) = CI G ( G ( q ) , −∇ Φ( · , λ + )) . In the proof we will apply the generalized equivariant Euler characteristic Υ G : [ GS ( ξ )] → U ( G ) defined for the category [ GS ( ξ )] of G -homotopy types of G -spectra, see [6] for defi-nition and properties. It is a natural extension of the equivariant Euler characteristic de-fined for finite, pointed G -CW-complexes. To prove this theorem it is sufficient to show that Υ G ( CI G ( G ( q ) , −∇ Φ( · , λ − ))) = Υ G ( CI G ( G ( q ) , −∇ Φ( · , λ + ))) . Let H = G q = Γ q × S and H = T ⊥ q G ( q ) ⊂ H π . Since CI H ( { q } , −∇ Ψ ± ) = CI H ( { } , −∇ ˜Ψ ± ) where ˜Ψ ± ( q ) = Ψ ± ( q + q ) and Ψ ± = Φ( · , λ ± ) | H : H → R , we study CI H ( { } , −∇ ˜Ψ ± ) . Set H = ker ∇ ˜Ψ ± (0) ⊕ im ∇ ˜Ψ ± (0) = N ull ⊕ Range where
N ull is a finite-dimensional, orthog-onal representation of H such that N ull S = N ull, i.e.
N ull ⊂ H consists of constant func-tions. Moreover, Range is an infinite-dimensional, orthogonal representation of H. Therefore if ω : N ull → Range is a H -equivariant map, then ω ( N ull ) ⊂ Range S = H ∩ Range, i.e. the im-age of ω also consists of constant functions. It is easy to prove Theorems 2.5.1 and 2.5.2 of [13] for H = G q = Γ q × S and with the assumption (F.3) of [13] replaced by N ull ⊂ H . To simplify no-tation, set A ± = ∇ ˜Ψ ± (0) | Range . Applying this slight modification of Theorem 2.5.2 of [13] we ob-tain ε > and H -equivariant gradient homotopies ∇H ± : ( B ε ( N ull ) × B ε ( Range )) × [0 , → H satisfying the following conditions:(1) ∇H ± (( u, v ) , t ) = A ± v − ∇ ξ ± (( u, v ) , t ) for t ∈ [0 , where ∇ ξ ± : H × [0 , → H is compactand H -equivariant such that ∇ ξ ± (0 , t ) = 0 , ∇ ξ ± (0 , t ) = 0 for any t ∈ [0 , , (2) ( ∇H ± ) − (0) ∩ ( B ε ( N ull ) × B ε ( Range )) × [0 ,
1] = { } × [0 , , i.e. is an isolated criticalpoint of ∇H ± ( · , t ) for any t ∈ [0 , , (3) ∇H ± (( u, v ) ,
0) = ∇ ˜Ψ ± ( u, v ) , (4) there exist H -equivariant, gradient mappings ∇ ϕ ± : B ε ( N ull ) → N ull such that ∇H ± (( u, v ) ,
1) = ( ∇ ϕ ± ( u ) , A ± v ) for all ( u, v ) ∈ B ε ( N ull ) × B ε ( Range ) , (5) ϕ ± ( u ) = e Ψ ± ( u, w ( u )) , ∇ ϕ λ ± ( u ) = P ∇ e Ψ λ ± ( u, w ( u )) is H -equivariant and P : H → N ull is the H -equivariant, orthogonal projection and moreover, since ω ( N ull ) ⊂ H , ϕ ± ( u ) = − πλ ± ˜ U ( u, w ( u )) where ˜ U : T ⊥ q Γ( q ) → R is defined by ˜ U ( q ) = U ( q + q ) . ERIODIC ORBITS 15
Since the H -equivariant Conley index of { } ⊂ H is invariant under the homotopy ∇H ± ( · , t ) , ≤ t ≤ and ∇H ± ( · , is a product map, we obtain the following CI H ( { } , −∇ e Ψ ± ) = CI H ( { } , −∇ ϕ ± ) ∧ CI H ( { } , − A ± ) ∈ [ HS ( ξ )] . (3.3.1)Since N ull ⊂ H is finite-dimensional, applying Remark 2.3.4 of [13], we have CI H ( { } , −∇ ϕ ± ) = CI H ( { } , −∇ ϕ ± ) , and so Υ H ( CI H ( { } , −∇ ϕ ± )) = χ H ( CI H ( { } , −∇ ϕ ± )) . (3.3.2)From the property (5) it follows that ∈ N ull is an isolated critical point of ϕ ± which is a localmaximum. Therefore, by Lemma 2.1.2, we obtain that χ H ( CI H ( { } , −∇ ϕ ± )) = χ H ( S Null ) ∈ U ( H ) . (3.3.3)Applying the generalized equivariant Euler characteristic Υ H to the equality (3.3.1) and com-bining the equalities (3.3.2) and (3.3.3) we get Υ H ( CI H ( { q } , −∇ Ψ ± )) = Υ H (cid:16) CI H ( { } , −∇ e Ψ ± ) (cid:17) == Υ H ( CI H ( { } , −∇ ϕ ± )) ∗ Υ H ( CI H ( { } , − A ± )) = χ H ( S Null ) ∗ Υ H ( CI H ( { q } , −∇ Π ± )) (3.3.4)where Π ± : Range → R is given by Π ± ( q ) = h∇ Ψ ± ( q ) | Range ( q − q ) , q − q i H π . Let Π n ± =Π ±| H n : H n → R and H n = ( T ⊥ q Γ( q ) ⊕ n M k =1 H k ) ∩ Range = (cid:16) T ⊥ q Γ( q ) ⊖ N ull (cid:17) ⊕ n M k =1 H k ⊂ Range ⊂ H . Analogously like in Lemma 3.3.2 of [13] it follows that there exists n ∈ N such thatfor any n ≥ n CI H ( { q } , −∇ Π n ± ) = CI H ( { q } , −∇ Π n ± ) (3.3.5)and CI H ( { q } , −∇ Π n ± ) = S H +1 , ± ⊕ W + (3.3.6)where the spectral decomposition of H n given by the isomorphism −∇ Π n λ ± is the following H n = H ⊕ ( T ⊥ q Γ( q ) ⊖ N ull ) ⊕ n M k =2 H k ! = ( H − , ± ⊕ H +1 , ± ) ⊕ ( W − ⊕ W + ) and dim H +1 , − = dim H +1 , + . Therefore the H -equivariant Conley indexes CI H ( { q } , −∇ Π − ) and CI H ( { q } , −∇ Π λ + ) are the H -homotopy types of H -spectra of the same type ξ = ( V n ) ∞ n =0 . Weput V n = V ⊕ V ⊕ · · · ⊕ V n . Now basing on the definition of the generalized equivariant Eulercharacteristic Υ H : [ HS ( ξ )] → U ( H ) we obtain that Υ H ( CI H ( { q } , −∇ Π ± )) = χ H (cid:16) S V n − (cid:17) − ⋆ χ H ( CI H ( { q } , −∇ Π n ± )) . (3.3.7)Recall that χ H ( S Null ) is an invertible element of U ( H ) , see Theorem 3.5 of [5]. Since N ull doesnot depend on the levels λ − and λ + , combining the equalities (3.3.4) and (3.3.7), we have thatthe condition Υ H ( CI H ( { q } , −∇ Ψ − )) = Υ H ( CI H ( { q } , −∇ Ψ + )) is equivalent to the condition CI H ( { q } , −∇ Π n − ) = CI H ( { q } , −∇ Π n + ) for any n ≥ n . From now on we consider two cases: Γ q ≈ T l or Γ is abelian. , ERNESTO PÉREZ-CHAVELA, AND SŁAWOMIR RYBICKI CASE: Γ q ≈ T l . For l = 0 we have proved this theorem in [13]. Let l > . In thiscase H ≈ T l × S . Analogously like in Lemma 3.3.2 of [13] we have H +1 , + = H +1 , − ⊕ U where U = { a cos t + b sin t : a, b ∈ V ∇ U ( q ) ( β j ) } and V ∇ U ( q ) ( β j ) is the eigenspace of ∇ U ( q ) corresponding to the eigenvalue β j . From Lemma 2.2.4 we get U H = { } , and consequentlythat U is a nontrivial representation of H . Thus CI H ( { q } , −∇ Π n − ) = S H +1 , − ⊕ W + , CI H ( { q } , −∇ Π n + ) = S H +1 , − ⊕ U ⊕ W + . (3.3.8)By the equalities (3.3.1) and (3.3.5) we obtain for any n ≥ n CI H ( { q } , −∇ Ψ n ± ) = S Null ∧ CI H ( { q } , −∇ Π n ± ) . According to Theorem 2.4.2 of [13] for isolated critical orbits we have CI G ( G ( q ) , −∇ Φ n ( · , λ ± )) = G + ∧ H CI H ( { q } , −∇ Ψ n ± ) for any n ≥ n . Therefore since U is a nontrivial representation of H , applying Theorem 2.2.3,we get for any n ≥ n χ G ( CI G ( G ( q ) , −∇ Φ n ( · , λ − ))) = χ G ( G + ∧ H S Null ⊕ H +1 , − ⊕ W + ) = χ G ( G + ∧ H S Null ⊕ H +1 , − ⊕ U ⊕ W + ) = χ G ( CI G ( G ( q ) , −∇ Φ n ( · , λ + ))) . Consequently CI G ( G ( q ) , −∇ Φ n ( · , λ − )) = CI G ( G ( q ) , −∇ Φ n ( · , λ + )) for any n ≥ n which im-plies that CI G ( G ( q ) , −∇ Φ( · , λ − )) = CI G ( G ( q ) , −∇ Φ( · , λ + )) . This completes the proof of thefirst case.3.3.2.
CASE: Γ is abelian. By Example 2.1.1, the pair ( G, H ) = (Γ × S , Γ q × S ) is admissible.To prove this case it is sufficient to show that Υ H ( CI H ( { q } , −∇ Ψ − )) = Υ H ( CI H ( { q } , −∇ Ψ λ + )) , see Theorem 2.4.3 of [13]. We have proved that this condition is equivalent to CI H ( { q } , −∇ Π n − ) = CI H ( { q } , −∇ Π n + ) for any n ≥ n . Let i ⋆ : U ( H ) → U ( S ) be the ring homomorphism induced by the inclu-sion i : { e } × S → H, see Remark 2.1.3. Like in the proof of the previous case we have CI H ( { q } , −∇ Π n − ) = S H +1 , − ⊕ W + and CI H ( { q } , −∇ Π n − ) = S H +1 , − ⊕ U ⊕ W + where U is a nontrivialrepresentation of S . Since U is a nontrivial representation of S and by the equality (3.3.5), forany n ≥ n we obtain the following i ⋆ ( χ H ( CI H ( { q } , −∇ Π n − ))) = i ⋆ ( χ H ( S H +1 , − ⊕ W + )) = χ S ( S H +1 , − ⊕ W + ) = = χ S ( S H +1 , − ⊕ U ⊕ W + ) = i ⋆ ( χ H ( S H +1 , − ⊕ U ⊕ W + )) = i ⋆ ( χ H ( CI H ( { q } , −∇ Π nλ + ))) , see Lemma 3.3.2 of [13]. Consequently CI H ( { q } , −∇ Π n − ) = CI H ( { q } , −∇ Π n + ) for any n ≥ n , which completes the proof of the second case. References [1] M. S. Berger. Bifurcation theory and the type numbers of Marston Morse. Proc. Nat. Acad. Sci. U.S.A.,69:1737-1738, 1972.[2] M. S. Berger. Nonlinearity and functional analysis: Lectures on nonlinear problems in mathematical analysis.Pure and Applied Mathematics. Academic Press, New York-London, 1977.[3] J. Fura, A. Ratajczak, and S. Rybicki. Existence and continuation of periodic solutions of autonomousNewtonian systems. J. Differential Equations, 218(1):216-252, 2005.
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