Symmetry Reduction by Lifting for Maps
SSymmetry Reduction by Lifting for Maps
H. R. Dullin , H.E. Lomel´ı , and J. D. Meiss ∗ School of Mathematics and StatisticsThe University of SydneySydney, NSW 2006, Australia
[email protected] Department of MathematicsInstituto Tecnol´ogico Aut´onomo de M´exicoMexico, DF 01000 [email protected] Department of Applied MathematicsUniversity of ColoradoBoulder, CO 80309-0526, USA
November 3, 2018
Abstract
We study diffeomorphisms that have one-parameter families of continuous symmetries.For general maps, in contrast to the symplectic case, existence of a symmetry no longerimplies existence of an invariant. Conversely, a map with an invariant need not have asymmetry. We show that when a symmetry flow has a global Poincar´e section there arecoordinates in which the map takes a reduced, skew-product form, and hence allows forreduction of dimensionality. We show that the reduction of a volume-preserving mapagain is volume preserving. Finally we sharpen the Noether theorem for symplecticmaps. A number of illustrative examples are discussed and the method is comparedwith traditional reduction techniques. ∗ HRD was supported in part by ARC grant DP110102001. HEL was supported by Asociaci´on Mexicanade Cultura and SNI 20216. JDM was supported in part by NSF grant DMS-0707659. a r X i v : . [ n li n . C D ] N ov Introduction
A symmetry group of a dynamical system may be discrete or continuous. The existence ofdiscrete symmetries implies the existence of related sets of orbits and imposes constraints onbifurcations [CL00, GS02]. Continuous symmetry often results in reduction; for example, theclassical results of Sophus Lie are concerned with the reduction of order of an ODE or PDEwith symmetry [Olv93]. In the Hamiltonian or Lagrangian context, a continuous symmetryimplies, through Noether’s theorem, the existence of an invariant, and the reduction of thedynamics by two dimensions [MR99, MMO +
07, HSS09].In this paper, we are are interested in global reduction theory for maps that have contin-uous symmetries. A continuous symmetry of a map f : M → M is a vector field whose flowcommutes with the map. The set of symmetries forms a Lie algebra. Symmetry reductionfor maps seems to have been first studied by Maeda [Mae80, Mae87], who showed that amap with an s -dimensional, Abelian Lie algebra of symmetries can be written locally in askew-product form F ( σ, τ ) = ( k ( σ ) , τ + ω ( σ )) , (1)where σ ∈ R n − s and τ ∈ R s , in the neighborhood of any point where the rank of thesymmetry group is s .We will show in § M (that is not necessarily compact), then we can find acovering map of the form p : Σ × R → M . If some topological conditions are satisfied, thenthe map f has a lift F to Σ × R with the skew-product form (1), see Thm. 3. The idearelies on the fact that the topologies of Σ and M may constitute an obstruction for this liftto exist. We call this procedure reduction by lifting ; it is a global version of Maeda’s result.This global reduction procedure is distinct from the general local reduction method dueto Palais [Pal61]. He showed that when an s -dimensional symmetry group has a properaction on M , then there is a codimension- s local neighborhood of each point, a Palais slice ,such that the group orbit of a slice forms a tube within which the orbit trivializes: thereare generalized “flow-box” coordinates. In this paper, we are primarily concerned withone-parameter symmetry groups, though our results can be extended to the case of multi-dimensional, Abelian groups. Moreover, we will show that the skew-product (1) can beglobally valid on M , or at least on an open, dense subset.In the classical theory of flows with Lie symmetry groups, a flow on a manifold M withsymmetry group G that acts properly on M , can be reduced to a flow on the space ofgroup orbits, M/G . When the action of G is free, then the group orbit space is a manifold;alternatively M/G is an “orbifold” and the reduction has singularities.By contrast, in our reduction procedure, there is no need to assume that the action of thesymmetry group is proper or free; it thus circumvents some of the problems with singularreduction. A number of examples are given in § § k of (1) is also volume preserving with respect to a natural volumeform on Σ. In particular, in the three-dimensional, volume-preserving case, the reduced map k : Σ → Σ is symplectic.As we recall in §
5, Noether’s theorem implies that whenever a Hamiltonian flow has aHamiltonian symmetry there is an invariant, and conversely, that every invariant generatesa Hamiltonian vector field that is a symmetry. This result also holds for symplectic twistmaps with a Lagrangian generating function [Log73, Mae81, WM97, Man06].We will generalize a result of Bazzani [Baz88] to show that, with the addition of arecurrence condition, Noether’s theorem also applies more generally to symplectic maps.However, it is important to note that symmetries do not generally lead to invariants nor doinvariants necessarily give rise to symmetries.In § In this section we investigate the conditions under which a map or flow that has a symmetrycan be written in the skew-product form (1). Whenever this is possible, the process reducesthe effective dimension of the dynamics by one. To accomplish this reduction, we supposethat the symmetry vector field generates a complete flow that admits a global Poincar´esection. We recall below the relationship between the existence of cross sections, coveringspaces and fundamental groups and show how to use it to obtain the symmetry reduction.
We start by recalling some basic notation (see also Appendix A) and facts about symmetriesand invariants of vector fields and maps. We suppose that M is a (not necessarily compact) n -dimensional, path-connected manifold and denote the set of C vector fields on M by V ( M )and the set of k -forms on M by Λ k ( M ). The flow, ϕ t ( x ), of a vector field X ∈ V ( M ) is thesolution of the initial value problem ˙ ϕ t ( x ) = X ( ϕ t ( x )) with ϕ ( x ) = x . In this paper we willassume that each vector field has a complete flow; that is, ϕ t : M → M is a diffeomorphismfor all t ∈ R . In this case, ϕ is an action of the one-dimensional Lie group R on M withcomposition rule ϕ t ◦ ϕ s = ϕ s + t .An invariant for a vector field X is a function I : M → R such that L X ( I ) = i X dI = 0 , where L X is the Lie derivative (see Appendix A). Equivalently I is invariant when ϕ ∗ t I = I .3imilarly I is an invariant for a map f : M → M if f ∗ I := I ◦ f = I . (2)A continuous symmetry of a vector field X is a vector field Y that commutes with X ,[ Y, X ] = 0, or equivalently, L Y X = 0. This implies that the flows of X and Y , say ϕ and ψ respectively, commute [Arn78]: ϕ t ◦ ψ s = ψ s ◦ ϕ t . Similarly, a vector field Y is a symmetry of a map f if f ∗ Y = Y , (3)where f ∗ Y := ( Df ( x )) − Y ( f ( x )). This means that the transformation f leaves the differen-tial equation ˙ y = Y ( y ) invariant. Since y and z = f ( y ) satisfy the same system of differentialequations, the symmetry extends to the Lie group generated by the flow of Y : ψ t ◦ f = f ◦ ψ t . Thus f is an equivariant transformation of the flow.Clearly, the collection of vector fields that are symmetries of a map form a Lie algebra.For example if X, Y ∈ V ( M ) are symmetries of f , then f ∗ [ X, Y ] = [ f ∗ X, f ∗ Y ] = [ X, Y ], so[
X, Y ] is also a symmetry.
We will assume f : M → M is a diffeomorphism with a symmetry vector field Y , (3).Standing assumptions are that M is an n -dimensional, path-connected manifold, and that Y has a complete flow with a global Poincar´e section.Recall that a relatively closed, codimension-one submanifold Σ (cid:44) → M is a Poincar´e section or cross section of a flow ψ if it is transverse to the flow, and is a global Poincar´e section of a complete flow if every orbit of the flow has both forward and backward transversalintersections with Σ, see the left pane of Fig. 1. Recall that Σ is relatively closed in M if,given a sequence in Σ that converges in M , the sequence also converges in Σ.When Σ is a global Poincar´e section for ψ , the first return time to Σ, T : Σ → R + , is thesmallest positive number such that for, each σ ∈ Σ, ψ T ( σ ) ( σ ) ∈ Σ. In fact, T is continuous.The first return map r ψ : Σ → Σ, also known as the Poincar´e map, is defined by r ψ ( σ ) := ψ T ( σ ) ( σ ) ; (4)it is a diffeomorphism. The sequence of return times T n , n ∈ Z , is defined so that T ( σ ) = 0, T n +1 ( σ ) := T n ( σ ) + T ( r nψ ( σ )) , n ≥ , and T − n ( σ ) = − T n ( r − nψ ( σ )). 4 ΣΣΣ a) b) ψ t ( σ ) ψ τ ( σ ) σ τ σ M C
Figure 1:
A global Poincar´e section Σ has two representations that are equivalent by Thm. 1: a ) as anembedding in the original manifold M , and b ) lifted to a cover C = Σ × R . In both cases, an orbit of theflow ψ is drawn. Using this notation, the iterates of the first return map are r nψ ( σ ) = ψ T n ( σ ) ( σ ) , (5)for all n ∈ Z . Note that the sequence of return times is strictly increasing with n , andmust be unbounded when the section Σ is relatively closed in M . Indeed, if for some σ thesequence T n ( σ ) were bounded then, since it is increasing, it would converge. Consequently,the difference T n +1 ( σ ) − T n ( σ ) → T ( r nψ ( σ )) → n → ∞ . Thus r nψ ( σ ) = ψ T n ( σ ) ( σ ) → σ ∗ would also converge, and since Σ is relativelyclosed, we could conclude that σ ∗ ∈ Σ. By continuity, T ( σ ∗ ) = 0, but this contradictsthe fact that the return times are always positive. A similar argument shows that T n ( σ ) isunbounded when n → −∞ .A global Poincar´e section Σ can be viewed both as a submanifold of M , and—as illus-trated in the right pane of Fig. 1—as the base of a covering space for M . Recall that amanifold C is a cover of M if there is a differentiable, surjective function p : C → M suchthat each m ∈ M has a neighborhood U for which p − ( U ) is the disjoint union of a countablenumber of open sets in C , each of which is diffeomorphic, via p , to U . The covering map p isnecessarily a local diffeomorphism. A covering space is a fiber bundle with a discrete fiber.We will often think of Σ both as a submanifold of M and a space in its own right.Technically, we define an inclusion map ι : Σ (cid:44) → M to express the embedding, but whenthere is little risk of confusion, we will let Σ ⊂ M denote ι (Σ).A fundamental tool for the rest of the paper is the following theorem of Schwartzmanthat shows how global Poincar´e sections are related to covering spaces.5 heorem 1 ([Sch62]) . A relatively closed submanifold Σ is a global Poincar´e section of aflow ψ on a manifold M if and only if the map p : Σ × R → M defined by p ( σ, τ ) = ψ τ ( σ ) (6) is a smooth cover of M with an infinite, cyclic group of deck transformations. Recall that a deck transformation of p is a map ∆ : C → C on the covering space suchthat p ◦ ∆ = p ; that is, deck transformations are the lifts of the identity [tD08]. For theglobal Poincar´e section Σ, is easy to verify that∆( σ, τ ) = ( r ψ ( σ ) , τ − T ( σ )) , (7)is a deck transformation on Σ × R . Taking into account (5), the iterates of ∆ satisfy∆ n ( σ, τ ) = ( r nψ ( σ ) , τ − T n ( σ )) (8)and are also deck transformations. Hence, the collection { ∆ n : n ∈ Z } , is the cyclic groupmentioned in Thm. 1. We next note that the orbits of ∆ correspond to the fibers of thecover. Lemma 2.
For each x ∈ M , the fiber, p − ( x ) , of the covering map p is an orbit of ∆ . Inother words, if p ( σ , τ ) = x , then p − ( x ) = orb ∆ ( σ , τ ) := { ∆ n ( σ , τ ) : n ∈ Z } . Proof.
For any point ( σ (cid:48) , τ (cid:48) ) ∈ orb ∆ ( σ , τ ), there is an n such that ( σ (cid:48) , τ (cid:48) ) = ∆ n ( σ , τ ), andsince ∆ n is a deck transformation, p ( σ (cid:48) , τ (cid:48) ) = p ◦ ∆ n ( σ , τ ) = p ( σ , τ ) = x . Conversely,suppose that ( σ (cid:48) , τ (cid:48) ) ∈ p − ( x ), i.e., ψ τ (cid:48) ( σ (cid:48) ) = x = ψ τ ( σ ) , so that σ (cid:48) = ψ τ − τ (cid:48) ( σ ) ∈ Σ.Assuming, without loss of generality, that τ > τ (cid:48) , then since the sequence T n ( σ ) is strictlyincreasing and unbounded, there must be an n ∈ N so that T n ( σ ) ≤ τ − τ (cid:48) < T n +1 ( σ ) . If we let σ n = r nψ ( σ ) and κ = τ − τ (cid:48) − T n ( σ ), then 0 ≤ κ < T ( σ n ) and σ (cid:48) = ψ κ ( σ n ). Bydefinition of the first return time, the only possibility is that κ = 0.Therefore, τ − τ (cid:48) = T n ( σ ). Consequently, σ (cid:48) = r nψ ( σ ) and thus ( σ (cid:48) , τ (cid:48) ) = ∆ n ( σ , τ ). We now ask whether it is possible to lift f to a cover C in which the map takes the skew-product form (1). Recall that a lift of f is a map F : C → C such that f ◦ p = p ◦ F . (9)6n particular, any deck transformation is the lift of the identity on the base. Lifts are notunique, but all lifts are the same up to deck-transformations: if G is another lift of f then G = ∆ m F for some integer m .Suppose that x , y ∈ C are two points in the cover such that f ( p ( x )) = p ( y ). Anecessary and sufficient condition [Mas77] for the existence of a lift is that f ∗ p ∗ π ( C, x ) ⊆ p ∗ π ( C, y ) , (10)where π ( C, x ) is the fundamental group of C based at x . In addition, if the lift exists, itcan be taken to satisfy F ( x ) = y . In other words, each loop in M that is a projectionof a noncontractable loop in C must map, under f , to another loop in the same projection.This is a nontrivial requirement because p ∗ ( π ( C, x )) is necessarily a subgroup of π ( M, x ).Note, however that the requirement would be trivially satisfied if the cover were simplyconnected, since in that case π ( C, x ) would be trivial. MMC Cp pF ? f δ p ° δ f ° p ° δγ γ Figure 2:
Failure of (10) in an attempt to construct a lift of the cat map to the cylinder.
The necessity of the condition (10) is easy to illustrate. Consider the example of Arnold’scat map on the two-torus M = R / Z : f ( x, y ) = (2 x + y, x + y ). Does f have a lift tothe cylinder C = R × S with the natural covering map, p ( z, τ ) = ( z mod 1 , τ )? Thefundamental group π ( M,
0) has two generators, say γ , γ ; however, π ( C,
0) has only onegenerator, say δ . Since p ∗ ( δ ) = γ , and f ∗ ( γ ) = γ + γ , the condition (10) is not satisfied.The geometric reason this fails is illustrated in Fig. 2. Any loop f ◦ p ◦ δ has the homotopytype of γ + γ . However, any lift of f ◦ p ◦ δ can not be a loop on C . Recall that the fundamental group of a topological space M depends on the choice of a base point ξ .However, the choice of base point is irrelevant provided the space M is path-connected.
7e will now give conditions for existence of a lift of f on the cover C = Σ × R with p : C → M as in (6). Let Ψ denote the trivial flow on Σ × R ,Ψ t ( σ, τ ) = ( σ, τ + t ) . (11)Clearly, Ψ commutes with ∆ and p ◦ Ψ t ( σ, τ ) = p ( σ, τ + t ) = ψ τ + t ( σ ) = ψ t ◦ p ( σ, τ ) so thatΨ is a lift of ψ , in the sense that each Ψ t is a lift of ψ t , for all t . Theorem 3 (Symmetry Reduction) . Suppose that a diffeomorphism f : M → M has asymmetry ψ t with global Poincar´e section Σ that is relatively closed in M and let p : Σ × R → M be the induced cover (6). Let σ ∈ Σ and (¯ σ, ¯ τ ) ∈ p − ( f ( σ )) .Then the following statements are equivalent.a) There exists a lift F : Σ × R → Σ × R of f to the cover of the form F : (cid:26) σ (cid:48) = k ( σ ) ,τ (cid:48) = τ + ω ( σ ) , (12) where k is a diffeomorphism of Σ , and k ( σ ) = ¯ σ , ω ( σ ) = ¯ τ .b) If ι : Σ (cid:44) → M is the standard inclusion, then f ∗ ι ∗ π (Σ , σ ) ⊆ p ∗ π (Σ × R , (¯ σ, ¯ τ )) . (13) Proof. a)= ⇒ b)By (12), F ( σ,
0) = ( k ( σ ) , ω ( σ )), and since F is a lift of f , f ( σ ) = f ◦ p ( σ,
0) = p ( k ( σ ) , ω ( σ )). Let G : Σ → Σ × R be given by G ( σ ) := ( k ( σ ) , ω ( σ )). Then p ◦ G = f ◦ ι and G ( σ ) = (¯ σ, ¯ τ ). Consequently f ∗ ι ∗ π (Σ , σ ) = p ∗ G ∗ π (Σ , σ ), and since G ∗ π (Σ , σ ) ⊆ π (Σ × R , (¯ σ, ¯ τ )), and p ∗ is injective, this directly implies b).b)= ⇒ a) Under the condition (13), standard theorems of algebraic topology [Mas77] implythat there exists a function G : Σ → Σ × R such that p ◦ G = f ◦ ι . Define k and ω by G ( σ ) =( k ( σ ) , ω ( σ )). Since f ◦ ι ( σ ) = p (¯ σ, ¯ τ ) = f ( σ ), we can take G ( σ ) = ( k ( σ ) , ω ( σ )) = (¯ σ, ¯ τ ).Now, for each τ ∈ R , we define F ( σ, τ ) := Ψ τ ( G ( σ )), where Ψ is the trivial flow (11).We claim that F is a lift of f , i.e., it satisfies (9). To show this, we notice that p ◦ F ( σ, τ ) = p ◦ Ψ τ ( G ( σ )) = ψ τ ◦ p ( G ( σ )) = ψ τ ( f ( σ )) = f ( ψ τ ( σ )) = f ◦ p ( σ, τ ) . The geometry underlying the reduction (12) is illustrated in Fig. 3. Theorem 3 can beseen to be equivalent to a commuting diagram. Let j : Σ → Σ × R be the inclusion given by j ( σ ) := ( σ, ι : Σ (cid:44) → M is the standard inclusion, then p ◦ j = ι . Let Π : Σ × R → Σ8 = ψ -τ ( x ) x x ' σ ' ff k τ τ ω( σ )ψ Σ Figure 3:
Showing the ideas behind the skew-product form (12). be the canonical projection. Clearly, Π ◦ j = id Σ . With this notation, Thm. 3 can berepresented by the commutative diagram Σ k (cid:47) (cid:47) ΣΣ j (cid:47) (cid:47) (cid:116)(cid:20) ι (cid:39) (cid:39) (cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78)(cid:78) id Σ (cid:55) (cid:55) (cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112) Σ × R Π (cid:79) (cid:79) F (cid:47) (cid:47) p (cid:15) (cid:15) Σ × R Π (cid:79) (cid:79) p (cid:15) (cid:15) M f (cid:47) (cid:47) M (14)In particular, the reduced map k on Σ can be written as k = Π ◦ F ◦ j . (15)Furthermore, the choice of σ and (¯ σ, ¯ τ ) such that f ( σ ) = ψ ¯ τ (¯ σ ), fixes the choice of k and ω . A simplification in Thm. 3 occurs if f has a fixed point at a point σ ∈ Σ. Since the inclu-sion j induces an isomorphism of the fundamental groups, j ∗ : π (Σ , σ ) → π (Σ × R , ( σ , ι ∗ π (Σ , σ ) = p ∗ j ∗ π (Σ , σ ) = p ∗ ( π (Σ × R , ( σ , . Upon taking ¯ τ = 0 and ¯ σ = σ , the condition (13) then reduces to f ∗ ι ∗ π (Σ , σ ) ⊆ ι ∗ π (Σ , σ ) . (16)Finally, we note a convenient equation determining k and ω is obtained by combiningthe definition of the lift (9), the form of the covering map (6), and the skew-product form(12) to obtain f ( ψ τ ( σ )) = ψ τ + ω ( σ ) ( k ( σ )) . Since f commutes with ψ τ we find k ( σ ) = ψ − ω ( σ ) ( f ( σ )) ∈ Σ . (17)9his determines k and ω (up to the choice of lift) by the requirement that the right sidemust be in Σ. In this section we will show that lifts of maps with symmetries have a homotopy invariant.
Lemma 4.
Suppose that f and its lift F satisfy the hypotheses of Thm. 3, and that ∆ is thedeck transformation defined in (7). Then there exists m ∈ Z such that F ◦ ∆ = ∆ m ◦ F andthe following identities are satisfied k ◦ r ψ = r mψ ◦ k ,ω ( σ ) − ω ( r ψ ( σ )) = T m ( k ( σ )) − T ( σ ) . The integer m is a homotopy invariant of the map f . In addition, if F is homotopic to theidentity then m = 1 .Proof. Consider the map G = F ◦ ∆ ◦ F − . Clearly p ◦ G = p , so G has to be a decktransformation. This implies that there exists m ∈ Z such that G = ∆ m .The integer m is independent of the choice of lift. Indeed, suppose that ˜ F is another liftof f , then since p ◦ F = p ◦ ˜ F , it follows from Lem. 2 that there is an integer j such that forany ( σ, τ ), ˜ F ( σ, τ ) = ∆ j ◦ F ( σ, τ ). By continuity, j is independent of ( σ, τ ), and consequentlyany two lifts differ at most by a deck transformation. Thus ˜ F ◦ ∆ = ∆ j + m F = ∆ m ˜ F .If F is homotopic to the identity then ∆ − ◦ G is a deck transformation that is homotopicto the identity. The only possibility is that ∆ − ◦ G is the identity and therefore m = 1.Similarly since the integer m is independent of the lift, continuity implies any two homotopicmaps will have the same value of m .For the rest, it is enough to use (8) in the skew product (12). Theorem 3 also applies to a flow ϕ , with a symmetry ψ . As in the theorem, we assume that ψ has a global Poincar´e section Σ. Corollary 5 (Symmetry Reduction of Flows) . Suppose that ψ, ϕ are a pair of commutingflows on M and that ψ has a global Poincar´e section Σ that is relatively closed in M . Thenthere is a lift Φ t of ϕ t to Σ × R of the form Φ t ( σ, τ ) = ( k t ( σ ) , τ + ω ( σ, t )) . (18) In addition, Φ t can be chosen to be a flow on Σ × R and hence the functions k t and ω ( · , t ) satisfy k t + s = k t ◦ k s ,ω ( σ, t + s ) = ω ( σ, t ) + ω ( k t ( σ ) , s ) . roof. Just as in the definition (6) of the cover p , let q : Σ × R → M denote the map q ( σ, t ) = ϕ t ( σ ). We notice that q ◦ j = p ◦ j = ι . Hence for each σ ∈ Σ, q ∗ π (Σ × R , ( σ , q ∗ j ∗ π (Σ , σ ) = p ∗ j ∗ π (Σ , σ ) = p ∗ π (Σ × R , ( σ , . Consequently, there exists a map Q : Σ × R → Σ × R such that q = p ◦ Q . Equivalently,there exist functions k ( σ, t ) and ω ( σ, t ) such that Q ( σ, t ) = ( k ( σ, t ) , ω ( σ, t )) and ϕ t ( σ ) = ψ ω ( σ,t ) ( k ( σ, t )). Denoting k t := k ( · , t ), define Φ t by (18). For each t , Φ t is clearly a lift of ϕ t ,since p (Φ t ( σ, τ )) = ψ τ + ω ( σ,t ) ( k ( σ, t )) = ψ τ ϕ t ( σ ) = ϕ t ψ τ ( σ ) = ϕ t ( p ( σ, τ )) . We must show that it is possible to choose Φ t so that Φ = id Σ × R , and Φ t + s = Φ t ◦ Φ s .Note first that Φ is a lift of the identity map and so it must be a deck transformation. If Φ is not itself the identity, then we can replace Φ t by Φ t ◦ Φ − . By Lem. 4, since Φ t is a mapthat is homotopic to the identity, it commutes with deck transformations; thus the new Φ t is still a lift of ϕ t , and satisfies Φ = id Σ × R .Now, for any s ∈ R , let G t = Φ t + s ◦ Φ − t ◦ Φ − s . Note that p ◦ G t = p , so that G t is adeck transformation. Moreover, since G = id Σ × R , G t is homotopic to the identity, and thusmust be the identity. In this way we conclude that Φ t satisfies the group property, and is aflow.The implication of Cor. 5 is that k t is a flow on the section Σ. Recall that generally the existence of a symmetry does not imply that of an associatedinvariant. However, it is possible that both the map and the symmetry do share an invariant.In that case, we can see that the reduced map has the same invariant.
Lemma 6.
Suppose f : M → M is a map with a symmetry ψ and the hypotheses of Thm. 3hold. If I : M → R is an invariant of f that is also an invariant of ψ , then ι ∗ I is aninvariant of k .Proof. Notice that F ∗ p ∗ I = p ∗ f ∗ I = p ∗ I. Moreover, for any ( σ, τ ) ∈ Σ × R , (6) implies p ∗ I ( σ, τ ) = I ( ψ τ ( σ )) = I ( ι ( σ )) = I ( ι (Π( σ, τ ))) = Π ∗ ι ∗ I ( σ, τ ) . Using the expression (15) we can then conclude that k ∗ ι ∗ I = j ∗ F ∗ Π ∗ ι ∗ I = j ∗ F ∗ p ∗ I = j ∗ p ∗ I = ι ∗ I .
Examples that have both invariants and symmetries will be given in § .7 Circle actions We have assumed that the symmetry vector field Y generates a flow ψ t that correspondsto an action of the Lie group R on the manifold M . However, in many cases, this actionis periodic, and can thus be thought of as an action of the group S on M . Typically, thetemporal period of the orbit of ψ t will depend upon the point. Nevertheless, if ψ t has aglobal section Σ, and every orbit is periodic, then some iterate of the Poincar´e map, r ψ , (4),must be the identity, because the flow returns to the original point on the same circle. Inthis case there is a smallest positive integer (cid:96) such that r (cid:96)ψ = id Σ , for all σ ∈ Σ.The existence of a global section for a circle action ψ t on M is a strong restriction on thetopology of the manifold. Indeed, since the (cid:96) -th iterate of the Poincar´e map is the identity,a Z (cid:96) ≡ Z \ (cid:96) Z covering of M is a trivial bundle over S , i.e., M = (Σ × S ) / Z (cid:96) .In general the existence of a (free and proper) circle action gives M the structure of aprincipal S -fiber bundle, though it need not be trivial. A classical example is provided bythe Hopf fibration of S . At first it seems as if the reduction of Thm. 3 could not be done fora symmetry that is a non-trivial bundle that is not a discrete quotient of a direct productwith S . However, one can often modify M in a way that it acquires the necessary topology.This may be achieved by removing parts of M that are invariant under the symmetry flowand under the map; examples are given § §
6. Some global topology may be lost becausethe modified M will in general not be compact.The question can also be inverted: how can one possibly achieve reduction and recon-struction in the case where the symmetry group action induces a non-trivial fiber bundle?The semi-direct product structure of (12) requires that the dynamics in the fiber are drivenby the dynamics in the base. However, if it is not possible to globally define an origin in the S fiber, this cannot be done. Classical symmetry reduction begins with a smooth manifold M (say without boundary, oreven better a compact manifold) and a Lie group G with a smooth and proper G -actionΦ : G × M → M . Two points on the same group orbit are now considered equivalent,and symmetry reduction means to study dynamics on the orbit space M/G . In general thisquotient is not a manifold, but just an orbifold. Each point x ∈ M has an associated isotropygroup, namely the set of elements of G that fixes x , G x = { g ∈ G : g ( x ) = x } . The conjugacy class of G x is called the orbit type and gives a stratification of M intotypes. This induces a stratification of the orbit space M/G with a smooth structure [Pfl01].When the group action is free (or slightly more generally, if the isotropy group is the samefor each point) the orbit space is a manifold, and M has the structure of a principal bundleover the orbit space. This is the standard setting for symmetry reduction. Reconstruction,12.e., the study of the dynamics in the fiber, may still be challenging globally because thefiber bundle may not be trivial: it may not have a global section.Many interesting group actions are not free. When the Lie group is compact, singularreduction is still possible, where “singular” reminds us that the reduced space in this caseis not a manifold. The standard tool is the Hilbert basis of invariants of the group action,which together with all relations and inequalities give an accurate description of the singularreduced space, see e.g. [Cho02, Pfl01]. We will use this approach in passing in some of theexamples in § § M [Pfl01, CL00, GSS88]. Moreover, its quotient by G is connected [Pfl01, Thm 4.3.2].In many cases, the isotropy group of the principal orbits is the trivial group. Within theprincipal stratum there is a subgroup of G that acts freely, so we return to the simpler caseof a free action, albeit on the smaller space obtained by removing certain closed subsetsfrom the original M . The fact that the symmetry commutes with the map f implies thatevery stratum of M is a forward invariant set of f . When f is invertible, this set is bothforward and backward invariant, but for non-invertible maps there may be points outsidethe invariant set that map into it.Hence when f is a diffeomorphism, the principal stratum is an open subset of full dimen-sion that is invariant under f . In the examples to follow, the “global Poincar´e section” ofthe symmetry is often taken to be a section on this principal stratum.For example, consider the circle action on C given by G = { ψ τ ( z , z ) = ( e ilτ z , e imτ z ) | ( z , z ) ∈ C , τ ∈ S } (19)for positive integers l and m . Note that if ψ t is a symmetry of a map f , then f must have,e.g., the invariant (cid:61) (¯ z m z l ), where (cid:61) ( z ) represents the imaginary part of z .When l = m = 1 the only non-trivial isotropy group is found at the origin. The principalstratum hence is the open set C \ { } , on which the action of ψ τ is free, so that it becomesa bundle with fiber S . Nevertheless, the principal stratum C \ { } does not have a globalPoincar´e section. This would imply that it is a fiber bundle with base S , which is impossiblesince every loop in C \ { } is contractible. The action may be restricted to the invarianthyperboloids given by the level sets (cid:61) ( z ¯ z ) = L . Whenever L (cid:54) = 0, the hyperboloid topolog-ically is R × S and there is a global section. Consequently upon removing the whole cone (cid:61) ( z ¯ z ) = 0 from C , each connected component does have a global section. Alternativelyone may remove one circle from each invariant S defined by | z | + | z | = const , whichturns S into R × S . Which “surgery” to chose depends on properties of the map f .When l = 1 and m > , z ) withdiscrete isotropy group t = j/m , j = 1 , . . . , m −
1. Every map with symmetry ψ τ has thisset as an invariant plane. Removing this plane gives the principal stratum A × C , where A = C \ { } . The section (cid:61) ( z ) = 0, (cid:60) ( z ) > z = 0 as an invariant boundary,and hence this section is a global Poincar´e section for maps with symmetry ψ . Thus we finda trivial bundle A × C with fiber S and base R + × C .13hen both l and m are bigger than 1 there are three strata: the origin (0 , , z ) and the plan ( z , C turns the manifold into A where A = C \ { } . A section along the positive real axis in either punctured plane A gives a globalPoincar´e section. In this section we will give several examples to illustrate Thm. 3. We start with a classicaltwo-dimensional example.
Example (M¨obius Map) An elliptic M¨obius transformation is conjugate to the form f ( z ) = az + b − bz + a , for real a, b with ab (cid:54) = 0. This map has fixed points at z = ± i and is analytic on theupper half plane M = { z ∈ C : (cid:61) ( z ) > } . As was noted by [Mae87], f has the symmetry Y = ( z + 1) ∂∂z , with flow ψ t ( z ) = cos( t ) z + sin( t )cos( t ) − sin( t ) z . On M the orbits of ψ are simply circles of radius r and center i √ r + 1 for any r ≥ z = i is a fixed point of ψ t .However on the principal stratum M \ { i } , the action is free, and has the global Poincar´esection Σ = { iσ : 0 < σ < } , with return time T = 2 π . Notice that in order to achieve aglobal Poincar´e section, the map the fixed point must be removed from the upper half plane.To compute the reduced form (12), we use the covering (6), p ( σ, τ ) = ψ τ ( σ ) and the notation(12), F ( σ,
0) = ( k ( σ ) , ω ( σ )) on the section, to obtain f ◦ p ( σ,
0) = f ( σ ) = aσ + ba − bσp ◦ F ( σ,
0) = ψ ω ( σ ) ( k ( σ )) = cos( ω ( σ )) k ( σ ) + sin( ω ( σ ))cos( ω ( σ )) − sin( ω ( σ )) k ( σ ) . Since, by (9), these must be equal we conclude that F takes the form F ( σ, τ ) = ( σ, τ + ω ( σ )) , where ω ( σ ) ≡ ω is, in fact, any constant such that ω = arg( a + ib ). The most natural choicemay be the principal value Arg( a + ib ) ∈ ( − π, π ], and each branch of arg gives a differentlift. Note that both f and Y have the invariant I ( z ) = 1 (cid:61) ( z ) ( | z | + 1) . According to Lem. 6, an invariant for k is naturally ι ∗ I = σ + σ , or trivially σ itself.14 xample (Twisted Symmetries): On the manifold M = R × ( R / Z ), define f ( ξ, z ) = ( R βz h ( R − βz ξ ) , z + α ( | ξ | )) , (20)where α : R + → R , R θ ∈ SO (2) is the rotation by angle 2 πθ , and β ∈ R . The map h : R → R is smooth and, so that f be continuous on M , is assumed to have the symmetry h ◦ R β = R β ◦ h . (21)Note that if h were to commute with R θ for all θ , then R βz h ( R − βz ξ ) = h ( ξ ), so (20) wouldalready be written in the skew-product form. Instead we assume that β is rational, so thatthe symmetry (21) is discrete.The map (20) has the symmetry ψ t ( ξ, z ) = ( R βt ξ, z + t ) . (22)This flow has a global Poincar´e section Σ = { ( σ,
0) : σ ∈ R } , with the covering map p ( σ, τ ) = ψ τ ( σ, F whichcan be easily computed using (17): F ( σ, τ ) = ( R − βα ( | σ | ) h ( σ ) , τ + α ( | σ | )) . (23)Since β = pq is rational, then by (21) h must have the associated discrete ( p, q ) symmetry. Inthis case, the orbits of ψ are ( p, q ) torus knots on M \ { ξ = 0 } . The action of the symmetrygroup is not free since the isotropy group of any point on the z -axis is different from thosenot on the axis. This example shows that even when the orbit space M/G is not a manifold,the reduced map of Thm. 3 may still be as smooth as the original map. The point is that(23) has a residual discrete symmetry with the action g ( ξ, τ ) = ( R β ξ, τ ). Factoring out thisdiscrete symmetry would indeed lead to a singular reduced space.An example of an orbit of the reduced map in the section Σ is shown in Fig. 4, and manyother examples of such maps can be constructed by replacing h with any map with a discretesymmetry like those in [FG09]. Example (Non-orientable manifold): For ( ξ, z ) ∈ R × R , let G be the discrete groupgenerated by g ( ξ, z ) = ( − ξ, z + 1). Note that G (cid:39) Z , and the action ( g, ( ξ, z )) (cid:55)→ g ( ξ, z )on R is free and proper. This implies that M = R /G is a manifold; in this case M isnon-orientable.Let f : R → R be the map f ( ξ, z ) = (cid:0) R ( z + β ) / h (cid:0) R − z/ ξ (cid:1) , z + β (cid:1) . (24)where (as in the previous example) R θ is the rotation by angle 2 πθ , h : R → R , and β ∈ R .Note that since R ( z +1) / = − R z/ , f ◦ g = g ◦ f , thus f can be thought of as a map on thequotient M . 15igure 4: Points on an attractor for the reduced map k in (23) for β = . Here, using the complex form u = x + iy , following [CG88], we took h ( u, ¯ u ) = ( λ + u ¯ u ) u + γ ¯ u . The figure shows the case λ = − . γ = 0 . α ( | σ | ) = − . It is easy to see that the flow ψ t ( ξ, z ) = ( R t/ ξ, z + t ) . is a symmetry of (24). Note that ψ t ◦ g = g ◦ ψ t so that ψ t defines a flow on M . Moreover,since ψ t +1 = g ◦ ψ t the orbits of the symmetry ψ are embedded circles in M , and ψ has theglobal Poincar´e section Σ = { ( σ,
0) : σ ∈ R } ⊂ M .
The return map for this case is simply r ψ ( σ ) = ψ ( σ ) = − σ . Computation of the reducedmap using (17) gives F ( σ, τ ) = ( h ( σ ) , τ + β ).Note that (24) is volume preserving on M whenever h is area preserving; for example, themap h ( x, y ) = (cid:18) y, ayy + 1 − x (cid:19) is area preserving for any a ∈ R . This map also has the integral J ( x, y ) = y x + x − ayx + y , i.e., one has J ◦ h = J . In this case the function I ( ξ, z ) = J (cid:0) R − z/ ξ (cid:1) . is an integral of f . Since it is also an integral of ψ t , I ◦ ψ t = I , Lem. 6 implies that h hasthe reduced integral ι ∗ I = J , as we already knew.16 xample (Nonhyperbolic Cat Maps): Suppose f is a diffeomorphism of the n -torus M = R n / Z n that fixes the origin, f (0) = 0, and is homotopic to the map b ( ξ ) = Bξ , where B ∈ SL ( n, Z ) is an n × n unimodular matrix.We will assume that B has a simple eigenvalue λ = 1 with right eigenvector v and lefteigenvector w . The vector field V = v · ∇ generates the flow ψ t ( ξ ) = ξ + t v , (25)In order that V be a symmetry of f , condition (3) requires that Df ( ξ ) v = v for all ξ ∈ M .For example, the map f ( ξ ) = Bξ + φ ( ξ ) v , (26)with φ (0) = 0, is of the assumed form. It has the symmetry ψ t if φ ( ξ + t v ) = φ ( ξ ), for all t ,so that Dφ ( ξ ) v = 0.Since w is a left eigenvector of B , the set Σ = { ξ ∈ M : w · ξ = 0 mod 1 } is a codimension-one subspace invariant under b and transverse to v . Indeed since B has integer coefficients,Σ (cid:44) → M is a codimension-one torus; an example is sketched in Fig. 5. Moreover, it is clearthat since Σ is transverse to v , it is a Poincar´e section for the flow (25).Since f and b are homotopic and both fix the origin, f ∗ = b ∗ , as maps on the fundamentalgroup π ( M, b , so the group ι ∗ π (Σ ,
0) is invariantunder b ∗ . Consequently, f ∗ ι ∗ π (Σ ,
0) = b ∗ ι ∗ π (Σ , ⊆ ι ∗ π (Σ , . Therefore, Thm. 3 and condition (16) imply that there exists a lift F of f to the cover Σ × R of the form of (12) with a covering map p : Σ × R → M of the form p ( σ, τ ) = σ + τ v . Tosimplify the computations we will use an equivalent cover, (cid:101) p : T n − × R → M given by (cid:101) p ( σ, τ ) = Sσ + τ v , where ( S | v ) ∈ SL ( n, Z ) and the columns of the n × ( n −
1) integer matrix S form a basisfor Σ. Consequently, B ( S | v ) = ( S | v ) (cid:18) ˆ B
00 1 (cid:19) (27)with ˆ B ∈ SL ( n − , Z ). In this case, a lift of the form (12) exists and must satisfy (9). Forexample, a lift of (26) is F ( σ, τ ) = ( ˆ Bσ, τ + ω ( σ )) , where ω ( σ ) = φ ( Sσ ).An explicit case of the form (26) for n = 3 is B = − , and φ ( x, y, z ) = g ( x − y, y − z ) (28)17 v Figure 5:
The global Poincar´e section Σ of the flow (25) for the matrix (28) is an embedded submanifoldof the three-torus M . for a function g : T → R such that g (0) = 0. Note that 1 is an eigenvalue of B with rightand left eigenvectors v = (1 , ,
1) and w = ( − , , − { ( x, y, z ) ∈ M : x − y + z = 0 mod 1 } , (29)shown in Fig. 5, is invariant under b and is a global Poincar´e section for the flow (25).Moreover, since Dφv = 0, the map f has symmetry (25).The fundamental group of Σ is generated by the loops η ( t ) = (2 t, t, t ) and η ( t ) = ( t, t, t ).Setting u i = ι ∗ [ η i ] for i = 1 ,
2, it is easy to verify that f ∗ u = u and f ∗ u = − u + 3 u .Consequently, Thm. 3 implies there exists a lift of f of the form (12). Indeed, the coveringmap (cid:101) p ( σ, τ ) = − − − σ σ τ , gives ( σ , σ , τ ) = ( x − y, y − z, − x + 3 y − z ), so that τ = 0 corresponds to ι (Σ). Uponcomputing ˆ B from (27), the lift takes the form (12) with k ( σ ) = (cid:18) − (cid:19) σ ,ω ( σ ) = g ( σ , σ ) . Since the eigenvalues of k are γ ± where γ is the golden mean, the reduced map is an Anosovdiffeomorphism.Note that the one cannot freely replace (29) with any global section of (25) and still satisfythe topological requirement (10). For example, the torus ˜Σ = { (0 , y, z ) ∈ M } is also a global18ection for (25). However, the map b ( ξ ) = Bξ takes the generators [(0 , t, , , t )] ofthe fundamental group of ˜Σ into [( t, , − t )] and [(0 , t, t )], violating (10). Example (Resonant Circle Action): The flow ψ τ ( z , z ) = ( e ilτ z , e imτ z ) of (19) corre-sponds to a circle action on M = C and is familiar from the study of resonant, coupledoscillators. As already noted in § , z ,
0) whenever l > , z ) whenever m > C with the symmetry (19) can be written (see, e.g., [GSS88]) z (cid:48) = f ( ρ ) z + f ( ρ )¯ z m − z l ,z (cid:48) = f ( ρ ) z + f ( ρ ) z m ¯ z l − . where the f j ( ρ ) are complex-valued functions of the real invariants of (19), namely ρ = z ¯ z , ρ = z ¯ z ,ρ = (cid:60) ( z m ¯ z l ) , ρ = (cid:61) ( z m ¯ z l ) . (30)These four invariants form a Hilbert basis : every function invariant under the action (19)is a function of { ρ , ρ , ρ , ρ } (see also § Df is invariant under thesymmetry ψ τ , and hence a function of the invariants.As a particular example consider( z (cid:48) , z (cid:48) ) = ( z − iεm ¯ z m − z l , z − iεlz m ¯ z l − ) . This map is, to lowest order in ε , the time- ε flow of the Hamiltonian H = ρ , and so isapproximately a four-dimensional symplectic map in ( x , x , y , y ), up to terms of order ε . The flow of this Hamiltonian has two invariants, namely H itself, and the Hamiltonianthat generates the symmetry, lρ + mρ . When ε (cid:28) m > l ≥
1, then since ψ t does not act freely on M = C ,there are points with nontrivial isotropy, in particular the set { (0 , z ) : z ∈ C } . Thisset is forward invariant under both ψ t and f , and removing it from M gives the manifold˜ M = A × C (cid:39) R + × S × C . On this manifold ψ t has the global Poincar´e sectionΣ = { ( x , z ) , x > , z ∈ C } ⊂ ˜ M .
Thus to get a global section, it is not necessary to restrict to the principal stratum, whichwould entail, when l >
1, the removal of the points ( z ,
0) as well. On the section Σ theseadditional points are fixed under f .The restriction of f to ˜ M has a lift on Σ × R determined by (17), i.e., by the requirement e − iω ( x − iεmx m − z l , z − iεlx m z l − ) ∈ Σ . ω is determined by requiring the first component to be real and positive, and F isgiven by k ( x , z ) = ( x | − iεmx m − z l | , e − iω ( z − iεlx m z l − )) ,ω ( x , z ) = arg(1 − iεmx m − z l ) . Note that even though the set z = 0 is forward invariant, since f is not invertible there arepoints that map into the invariant set. Similarly, the reduced map k has points that mapinto the boundary x = 0 given by the solutions of z l = ( iεmx m − ) − . The reduced map k can be extended to the excluded set x = 0 by continuity, so that the pre-images of x = 0have well defined orbits. Even so, the fibre map ω is undefined for points that map into theset x = 0 so, strictly speaking, reduction by lifting fails in this case. However, even thoughthe fibre map is undefined for certain points, the reduced map k is well behaved. We will now show that, when the map f and its symmetry are both volume preserving, thereduced map k of (12) is also volume preserving on Σ, with respect to an appropriate volumeform. This specializes Thm. 3 to the volume-preserving setting. We denote the volume formby Ω ∈ Λ n ( M ); by assumption, both f and ψ t preserve this form, f ∗ Ω = Ω and ψ ∗ t Ω = Ω,respectively. Equivalently, the symmetry vector field is incompressible: L Y Ω = ( ∇· Y )Ω = 0,where ∇ · Y is the divergence of Y . When ∇ · Y = 0, we will say that Y is incompressible. Theorem 7.
In addition to the hypotheses of Thm. 3, assume that f is volume preservingand its symmetry Y is incompressible. Then the reduced map k of (12) preserves a volumeform ν on Σ defined by ν = ι ∗ i Y Ω . (31) Proof.
First we note that µ := i Y Ω is an ( n − M . Moreover, since Σ is a section, Y (cid:54)∈ T Σ, and so ν = ι ∗ µ is non-degenerate on Σ. In addition, dν = ι ∗ di Y Ω = ι ∗ ( L Y Ω − i Y d Ω) = 0 ; thus ν is a volume form. Finally, from (3) we have f ∗ µ = i f ∗ Y f ∗ Ω = i Y Ω = µ , so f preserves µ .We now assert that, in fact, Π ∗ ν = p ∗ µ , (32)where Π : Σ × R → Σ is the canonical projection. To see this recall that since p is a localdiffeomorphism and Y (cid:54) = 0 everywhere, there exists a flow box [AM78] in M near each pointof Σ. Therefore, we can reduce the proof to the case in which Σ is an open set in R n − , M isof the form M = Σ × R and p is the identity. If we let ( σ, τ ) be the coordinates of M then,by construction, we have that Y = ∂/∂τ and the flow on M is ψ t ( σ, τ ) = ( σ, τ + t ). In thesecoordinates, we have that ι ◦ Π( σ, τ ) = ( σ, κ ( σ, τ ) dσ ∧ · · · ∧ dσ n − ∧ dτ . ψ t is volume preserving, we conclude that κ ( σ, τ ) = κ ( σ, σ, τ ) ∈ M . Thisimplies that µ = i Y Ω does not depend on τ and therefore p ∗ µ = ( ι ◦ Π) ∗ µ = Π ∗ ν , which isequality (32).Finally from the definition (15) of k , and the fact that Π ◦ j = id Σ , we get that k ∗ ν = j ∗ F ∗ Π ∗ ν = j ∗ F ∗ p ∗ µ = j ∗ p ∗ f ∗ µ = j ∗ p ∗ µ = j ∗ Π ∗ ν = ν . In this way, we conclude that k preserves ν .Theorem 7 can be combined with Cor. 5 to show that ν is a reduced volume form for anincompressible vector field with an incompressible symmetry. Corollary 8.
Suppose that
X, Y ∈ V ( M ) are incompressible, commuting vector fields andthat Y has a flow ψ with a global Poincar´e section Σ that is an immersed manifold ι : Σ (cid:44) → M .Then there exists a vector field K on Σ and a function ζ : Σ → R , such that X | Σ = K + ζ Y | Σ , (33) where K is incompressible with respect to the volume form ν = ι ∗ i Y Ω on Σ .Proof. By Thm. 3 the flow ϕ of X has a lift Φ to the cover Σ × R of the form (18). Let K ∈ V (Σ) denote the vector field generated by the reduced flow k of Φ. By Thm. 7, each k t preserves the volume-form ν so that K is incompressible on Σ with respect to the form in(31).Finally, since K is tangent to—and Y is transverse to—Σ, the vector field X | Σ can bewritten as a linear combination of K and Y . Indeed (18) gives ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Φ t ( σ, τ ) = K + ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ω ( σ, t ) ∂∂τ . Since Y = ∂∂τ in the coordinates ( σ, τ ), this implies (33) with ζ ( σ ) = ˙ ω ( σ, M = R and noted that the flow on the two-dimensional group orbit spaceis Hamiltonian. Example (Volume-Preserving normal form):Consider the family of volume-preserving diffeomorphisms, f : T → T , defined by f ( x, y, z ) = ( x + a ( y ) , y + z + b ( y ) , z + b ( y )) . (34)Similar maps on R , arise as the normal form for a volume-preserving map near saddle-nodebifurcation with a triple-one multiplier [DM08]. The map (34) has symmetry Y = ∂∂x whichgenerates the flow ψ t ( x, y, z ) = ( x + t, y, z ). The natural section for ψ is Σ = { (0 , v, w ) : v, w ∈ T } , and the corresponding covering map p : T × R → T defined by p ( v, w, τ ) = ψ τ (0 , v, w ).21he group ι ∗ π (Σ ,
0) is generated by the equivalence classes γ = [(0 , t, γ = [(0 , , t )].Since a and b are a periodic functions, f ∗ ( γ ) = γ , f ∗ ( γ ) = γ + γ , If we assume that a (0) = b (0) = 0, f fixes the origin, and then the requirement (16) holds.Of course (34) is already in skew-product form, with τ = x , and ω ( v, w ) = a ( v ), and thereduced map becomes k ( v, w ) = ( v + w + q ( v ) , w + q ( v )). The latter is a generalized Chirikovstandard map and is typically chaotic.Notice that if a ( y ) = y the topological condition would fail for any choice of section. Oneway to see this is to note that the linear part of the map would then be a Jordan block thatdoes not have any 2-dimensional invariant subspaces. Another is to note that lifting withthe natural section would give ω ( u, v ) = u , which is not a continuous function from T → R .Finally, one might think that this problem could be repaired by first lifting f to the universalcover R , eliminating the topological requirement. However, in this case almost all orbits ofthe linear map are unbounded— there is no return map to any surface. Example (4D Volume-preserving map): Consider a map f = T ◦ T of R (cid:39) C withcoordinates ( z , z , ¯ z , ¯ z ) given by the composition of the two “shears” T ( z , z ) = (cid:16) z ( a + i (cid:112) c − a + h ( z ¯ z ) / ( z ¯ z )) , z (cid:17) T ( z , z ) = (cid:16) z , z ( b + i (cid:112) c − b + h ( z ¯ z ) / ( z ¯ z )) (cid:17) , where a, b, c , c are real constants, and h and h are non-negative real functions. Eachof the shears has constant Jacobian c i , so det Df = c c . Thus if c c = 1, f is volumepreserving. If c = c = 1 the map f is the composition of two symplectic maps.Each shear commutes with the two symmetries ψ t ( z i ) = e it z i . Taken together this im-plies that f has a pair of commuting symmetries, i.e., is equivariant under the T action ψ ( t ,t ) ( z , z ) = ( e it z , e it z ). The reduction of Thm. 7 can be carried out recursively, thusreducing f to an area-preserving map.To construct a global section, we must remove the sets of nontrivial isotropy from M : re-moving the fixed sets z = 0 and z = 0 leaves A = ( C \ { } ) . On this set there is a globalPoincar´e section for the two-parameter action, Σ = { ( x , x ) : x i > } ⊂ A . The result ofthe reduction is to simply introduce polar coordinates in both complex planes, and give afiber map with − ω = (arg( z (cid:48) ) , arg( z (cid:48) )).The reduced map k would be written in terms of ( x , x ) ∈ Σ. Instead we use the twoinvariant radii ρ = z ¯ z and ρ = z ¯ z as coordinates we find T ( ρ , ρ ) = ( c ρ + h ( ρ )) , ρ ) = ( ρ (cid:48) , ρ )and similarly for T , so that the reduced area-preserving map (assuming c c = 1) is T ( T ( ρ , ρ )) = ( c ρ + h ( ρ ) , c ρ + h ( ρ (cid:48) )) , which is smooth even for ρ j = 0. 22 Noether symmetries
For Hamiltonian flows and symplectic maps the close relationship between the existence ofinvariants and symmetries is well-known. Noether’s theorem, for example, implies that aLagrangian system with a symmetry acting in configuration space has an invariant. Thisrelationship is exploited in the Liouville-Arnold construction of action-angle variables for anintegrable n -degree-of-freedom Hamiltonian where each invariant generates a symmetry vec-tor field. In this case, the involutive property of the invariants implies that the correspondingsymmetry vector fields commute.As we remarked in the introduction, analogues of Noether’s theorem do not exist for mapsin general nor for volume-preserving maps in particular. For example the skew product f ( x, y, τ ) = ( k ( x, y ) , τ + ω ( x, y ))on Σ × S has the obvious symmetry Y = ∂/∂τ and preserves the volume form Ω = dx ∧ dy ∧ dτ whenever k is area preserving. If k has no invariant, then neither does f . This would occur,e.g., when Σ = T , and k ( x, y ) = (2 x + y, x + y ), the cat map.In addition, a three-dimensional map could have an invariant, restricting its orbits totwo-dimensional level sets, and yet have no symmetry: the map can be chaotic on the levelsets. Examples include the trace maps [RB94], such as the Fibonacci map f ( x, y, z ) = ( y, z, − x + 2 yz ) . Trace maps are volume preserving; many similar examples have also been constructed[GM02]. Consequently, symmetries do not necessarily give rise to invariants without someadditional structure.One sufficient additional structure corresponds to symplectic maps with Hamiltoniansymmetries. A symmetry Y that is a Hamiltonian vector field is known as a Noether sym-metry . Here we give a slight generalization of a theorem of [Baz88, App. 1], which requiredthe symplectic map to be a twist map. We show that one can drop this requirement if themap has a mild recurrence property.
Theorem 9 (Symplectic Noether) . If Y is a Noether symmetry of a symplectic map f and f has any recurrent orbits, then f has an invariant. Conversely if I is an invariant of asymplectic map f , then its Hamiltonian vector field is a symmetry of f .Proof. For this proof let µ denote the symplectic form (e.g., µ = dq ∧ dp ), and let I bethe Hamiltonian of the vector field Y : i Y µ = dI . Consequently f ∗ i Y µ = d ( f ∗ I ), but bythe symmetry and symplectic conditions f ∗ i Y µ = i f ∗ Y f ∗ µ = i Y µ = dI . Consequently d ( f ∗ I − I ) = 0, so that I ( f ( x )) = I ( x ) + c for some constant c . Suppose x ∗ is a recurrentpoint, then there exists a subsequence t i → ∞ such that f t i ( x ∗ ) → x ∗ . However, since I ( f t ( x )) = I ( f t − ( x )) + c = . . . = I ( x ) + tc , this implies thatlim i →∞ [ I ( f t i ( x ∗ ) − I ( x ∗ )] = lim i →∞ t i c = 0 ,
23o that c = 0. Thus I is an invariant for f . Conversely, if the symplectic map f has aninvariant I , then it generates a Hamiltonian vector field Y , and i f ∗ Y µ = f ∗ ( i Y µ ) = f ∗ ( dI ) = d ( f ∗ I ) = dI = i Y µ . Thus i f ∗ Y − Y µ = 0, but since µ is non-degenerate, this can only occur when f ∗ Y = Y .The condition that f has a recurrent orbit is necessary. The map f ( x, y ) = ( x + c, y + g ( x )) , on R is symplectic with the canonical form dx ∧ dy and has the translation symmetry Y = ∂/∂y . This symmetry is generated by the Hamiltonian I ( x, y ) = x , which, however, isnot an invariant for f for any c (cid:54) = 0. Note that f has no recurrent orbits. By contrast, if wetake the same f but suppose that it is a map on S × R , then f may have recurrent orbits,but the symmetry vector field ∂/∂y is locally, but not globally Hamiltonian, and so does notgenerate an invariant. Example (Lyness Map): For any a >
0, the Lyness map [Lyn45], f ( x, y ) = (cid:18) y, a + yx (cid:19) (35)is a diffeomorphism on the positive quadrant M = R that preserves the symplectic form µ = xy dx ∧ dy . It has the symmetry [BC98] Y = xy ( − ∂ y I, ∂ x I ) = − xy ∂I∂y ∂∂x + xy ∂I∂x ∂∂y , where I = (1 + x )(1 + y )( a + x + y ) xy . The function I is an invariant for f . Moreover, Y is the Hamiltonian vector field of I withrespect to µ and so I is an invariant for Y . This is in agreement with Thm. 9.The level sets of the invariant I for I > I min = I ( x ∗ , x ∗ ) are topologically circles that intersectthe line Σ = { ( σ, σ ) : σ > x ∗ } with x ∗ = (1 + √ a ) exactly once. The symmetry is notfree since the fixed point ( x ∗ , x ∗ ) has nontrivial isotropy. However on the principal stratum˜ M = M \ { ( x ∗ , x ∗ ) } , Σ is a global Poincar´e section for the flow of Y .Given that Σ is simply connected, f can be reduced with a lift. The lifted map on Σ × R is F ( σ, τ ) = ( σ, τ + ω ( σ )). An integral expression for ω ( σ ) can be obtained from the results in[BC98].In the next section, we will apply these results to a four-dimensional symplectic map.24 Comparison of reduction methods
In this section, we will illustrate three different approaches to reduction, namely • invariants, • polar coordinates, and • reduction by lifting.We will use, as an example, a four-dimensional symplectic map with a rotational symmetry.A symplectic map on M = R (cid:39) C of standard type with a symmetry can be constructedusing the Lagrangian generating function S ( z, z (cid:48) ) = | z − z (cid:48) | − W ( z ¯ z ) with z = x + iy ∈ C and W : R → R a C − function. This generating function is invariant under the rotation( e iτ z, e iτ z (cid:48) ). Introducing the conjugate momenta w = p x + ip y , then the corresponding 4Dmap, generated by w = − ∂ ¯ z S and w (cid:48) = ∂ ¯ z (cid:48) S , is f ( z, w ) = ( z (cid:48) , w (cid:48) ) = ( z + w (cid:48) , w − zV ( z ¯ z )) (36)where V ( x ) = ddx W ( x ). The map (36) has the symmetry ψ t ( z, w ) = ( e it z, e it w ), e.g., the flow(19) with k = l = 1. Note that ψ t is symplectic and is the flow of the “angular momentum”Hamiltonian, L = (cid:61) (¯ zw ). The discrete Noether theorem, Thm. 9, implies that L is aninvariant of f as well. Reduction using invariants:
Since the symmetry ψ t is a special case of (19), the basis(30) with ( z , z ) → ( z, w ) and k = l = 1 generates its real polynomial invariants. Note thatthese ρ i are not independent: they obey the relation ρ + ρ = ρ ρ , (37)and the inequalities ρ ≥ , ρ ≥ . The map ( z, w ) → ( ρ , ρ , ρ , ρ ) is called the Hilbert map . Rewriting (36) in terms of theinvariants gives ρ (cid:48) = ρ (1 − V ) + ρ + 2 ρ (1 − V ) ,ρ (cid:48) = ρ − ρ V + ρ V ,ρ (cid:48) = ρ − ρ V + ρ (cid:48) , (38)where V = V ( ρ ) and ρ = − L , the negative of the constant angular momentum. This isvolume preserving with respect to the form Ω = dρ ∧ dρ ∧ dρ on R [ ρ , ρ , ρ ] and has aninvariant ρ = ρ ρ − ρ inherited from the relation (37).The Hilbert coordinates satisfy the Poisson structure given by { ρ , ρ } = 4 ρ , { ρ , ρ } =2 ρ , { ρ , ρ } = − ρ with Casimir ρ ρ − ρ . Moreover, the reduced mapping (38) is a25oisson map: it preserves this reduced Poisson structure. The level set of the Casimir is asmooth submanifold provided ρ (cid:54) = 0. On each such a level set (a smooth symplectic leaf)the map is area preserving. The reduction by the symmetry and by its generating invariantreduces the dimension of the map by two.However, the invariant set ρ = 0 is a half-cone. This case is a singular reduction, becausethe action ψ t is not free, since the origin ( z, w ) = (0 ,
0) is a fixed point for the whole group.From this construction, the map in the symmetry direction, in the form τ (cid:48) = τ + ω ( ρ i ),can be obtained as follows. The Hilbert map from full phase space T ∗ R to the invariants isnot invertible. It has the symmetry orbits as fibers. As pre-image in the fiber we can choose( z, w ) = h ( ρ ) =: (cid:0) √ ρ , √ ρ e i arg( ρ − iρ ) (cid:1) . The angle increment ω is now defined by f ( h ( ρ )) = ψ ω h ( ρ (cid:48) ). The explicit form of thefiber map therefore is found from z (cid:48) = e iω ρ (cid:48) so that ω = arg( z (cid:48) ). The map h is not definedwhen ρ = ρ = 0, i.e. at the origin of the z -plane and at the origin of the w -plane. Reduction using polar coordinates:
Polar coordinates appear whenever the symme-try angle θ becomes a coordinate. There is a unique extension of this transformationof coordinates to a symplectic transformation of all of phase space that is linear in themomenta, the so-called cotangent lift, see, e.g. [AM78]. Denoting the new variables by( r, θ, p r , p θ ), the momentum p θ , conjugate to θ , becomes the conserved quantity. Specifi-cally set z = re iθ with cotangent lift w = e iθ ( p r − ip θ /r ) and rewrite the map in the newcoordinates ( r, θ, p r ) = ( √ z ¯ z, arg z, (cid:60) ( z ¯ w ) /r ).Having done most of the work in for the Hilbert basis already, we proceed as follows. Interms of the invariants, the polar coordinates are ( r, p r , p θ ) = ( √ ρ , ρ / √ ρ , ρ ) and the mapis obtained from that of the invariants by eliminating ρ using ρ = ( ρ + ρ ) /ρ = p r + p θ /r .This gives an area-preserving map that depends on the parameter p θ r (cid:48) = (cid:0) p θ /r + ( r + p r − rV ) (cid:1) / ,p (cid:48) r = (cid:0) r (cid:48) − r ( r + p r − rV ) (cid:1) /r (cid:48) , (39)where V = V ( r ). The fiber map is obtained from rewriting z (cid:48) = z + w (cid:48) = z + w − zV inpolar coordinates, hence r (cid:48) e iθ (cid:48) = e iθ ( r + p r − ip θ /r − rV ) and therefore θ (cid:48) = θ + arg( r + p r − rV − ip θ /r ) . This map works well unless r (cid:48) = 0, which makes p (cid:48) r and θ (cid:48) undefined. Points with r (cid:48) = 0 canonly be reached when p θ = 0, but in that case it happens as soon as r + p r = rV . Excluding p θ = 0 gives a well-defined, smooth map. Reduction by lifting:
For reduction by lifting we need to find a global Poincar´e sectionof the symmetry flow ψ t . As a first attempt try the surface defined by (cid:61) ( z ) = 0 for x =26 ( z ) >
0. The flow is tangent to the section when (cid:60) ( z ) = 0, hence all of the plane { z = 0 } is tangent to the section. Unfortunately, this plane of tangency is not invariant under themap. In order to avoid this difficulty, we can choose a larger invariant set that contains thetangency set, and restrict the map to the complement of this set.The angular momentum L ( z, w ) = −(cid:61) ( z ¯ w ) = (cid:61) (¯ zw ) is invariant both under the flow andthe symplectic map. Notice that the plane { z = 0 } is contained in the level set { L ( z, w ) = 0 } ,that is invariant. The manifold M \ { L ( z, w ) = 0 } has two connected components,˜ M = { ( z, w ) ∈ C × C : L ( z, w ) > } , and the corresponding negative angular momentum set. As the analysis for either componentis the same, we will now restrict the map f and the flow ψ t to ˜ M . The symmetry flow as aglobal Poincar´e section on ˜ M : Σ = { ( x, w ) ∈ ˜ M : x > } . This section is relatively closed in ˜ M and if ( x, w ) ∈ Σ then (cid:61) ( w ) > f is of the form F ( σ, τ ) = ( k ( σ ) , τ + ω ( σ )). Taking a point σ = ( x, w ) ∈ Σ we can compute its image under (36) to obtain f ( x, w ) = ( x + w − xV ( x ) , w − xV ( x )) . (40)By (17), this equals ψ ω ( σ ) ( k ( σ )), so that e − iω ( x + w − xV ) must be real and positive (where x is assumed positive). Noticing that (cid:61) ( x + w − xV ) >
0, this immediately gives ω ( σ ) =arg( x + w − xV ), and we can choose the branch so that 0 < ω ( σ ) < π . The reduced mapthus becomes k ( x, w ) = ( | x + w − xV | , e − iω ( x,w ) ( w − xV )) . By (31), k preserves the volume form ν = p y dx ∧ dp x ∧ dp y .Because we started with a symplectic map the reduced map has an invariant, ι ∗ (cid:61) (¯ zw ) = x (cid:61) ( w ) = xp y = p θ . If we were to eliminate p y by fixing the invariant, we would again obtainthe polar coordinate map (39). Comparison:
The Hilbert map is similar to what we call the projection Π to the sectionΣ, while its “inverse” h is like the inclusion map ι . In differential geometry, a fiber bundle issaid to have a “global section” if h is continuous. Notice, however, that the Hilbert map hasfiber S , while when we construct the global Poincar´e section we have a bundle with base S and fiber Σ. In fact, as pointed out in § S . In the present example the manifold ˜ M is topologically a direct product S × R × R + .The Hilbert map may be applied to points with isotropy where projection to Σ is notdefined. The reduced map (38) is well-defined, even when the S -bundle does not possessa global section or when there is no global Poincar´e section. Reduction by lifting is moreefficient in that one does not need to introduce extra coordinates; moreover, one does notneed to know the “good” coordinate system to do the reduction. Instead the choice of globalPoincar´e section defines such a coordinate system.27 Conclusions
We have studied maps with continuous symmetries, specializing to the case that the sym-metries have a global Poincar´e section. We showed in Thm. 3 that if a necessary topologicalcondition is satisfied, then the map has a lift such that in certain coordinates the lift hasthe skew-product form (12). We called this “reduction by lifting” because the map on thesection, the reduced map k , describes the dynamics modulo the symmetry. The fiber map,which corresponds to a translation, is obtained naturally from the symmetry flow.The topological conditions for the existence of a lift require that the homotopy homomor-phism induced by the map leaves the fundamental group of the Poincar´e section invariant.This requirement is trivial if the section is simply connected. In principle, the restrictioncould be avoided if one first lifts the dynamics to the universal cover; however—as we sawin the examples—almost all orbits may become unbounded. In many cases, a minimal liftusing only the symmetry flow seems more useful.If the map and its symmetry are volume-preserving, then the reduced map is also volume-preserving as shown in Thm. 7. If both the map and its symmetry preserve an invariant, thenthe reduced map also has an invariant. Finally, if the map and its symmetry are symplectic,then, as in Noether’s theorem, the existence of a symmetry implies an invariant, providingthe map has recurrent orbits, recall Thm. 9.A number of examples were given to compare and contrast reduction by lifting to stan-dard reduction techniques. Reduction by lifting is more parsimonious than the Hilbertmapping technique, which can result in a high-dimensional map that must satisfy a numberof constraints. It is more explicit than the standard reduction by group orbit technique, andmoreover applies when the action of the symmetry is neither proper nor free. When theaction is not free, reduction by lifting need not lead to a map on a singular space.However, the existence of a global section for the symmetry flow is a strong restriction.This can be overcome in some cases, as we showed, by restriction of the dynamics to invariantstrata of the symmetry flow. A Forms and Lie Derivatives
Here we set out our notation, which follows, e.g., [AMR88]. The set of k -forms on a manifold M is denoted by Λ k ( M ), and the set of C vector fields by V ( M ). If α ∈ Λ k ( M ) and V , V , . . . V k ∈ V ( M ), then the pullback, f ∗ , of a form α by a diffeomorphism f is defined by( f ∗ α ) x ( V , V , ..., V k ) = α f ( x ) ( Df ( x ) V ( x ) , . . . , Df ( x ) V k ( x )) . (41)The pullback can be applied to a vector field V as well:( f ∗ V )( x ) := ( Df ( x )) − V ( f ( x )) . (42)The push-forward operator is defined as f ∗ = ( f − ) ∗ . The inner product of α with V isdefined as the ( k − i V α := α ( V, · , . . . , · ) . ϕ t is the ( C ) flow of a vector field V , so that ϕ ( x ) = x , and ddt ϕ t ( x ) = V ( ϕ t ( x )). Then the Lie derivative with respect to V is the linear operator defined by L V ( · ) := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ ∗ t ( · ) (43)where · is any tensor. In particular for a vector field X , L V X = [ V, X ] (44)is the Lie bracket. The Lie derivative acting on differential forms obeys Cartan’s homotopyformula L V α = i V ( dα ) + d ( i V α ) . (45) References [AM78] R. Abraham and J.E. Marsden.
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