Positive-Entropy Integrable Systems and the Toda Lattice, II
aa r X i v : . [ n li n . S I] J u l POSITIVE-ENTROPY INTEGRABLE SYSTEMS AND THE TODALATTICE, II
LEO T. BUTLER
Abstract.
This note constructs completely integrable convex Hamiltonianson the cotangent bundle of certain T k bundles over T l . A central role is playedby the Lax representation of a Bogoyavlenskij-Toda lattice. The classifica-tion of these systems, up to iso-energetic topological conjugacy, is related tothe classification of abelian groups of Anosov toral automorphisms by theirtopological entropy function. Introduction
Say that a smooth flow ϕ : M × R → M is integrable if there is an open densesubset L ⊂ M such that L is fibred by b -dimensional tori and the smooth bundlecoordinate charts ( I, φ ) : U → D a × T b conjugate ϕ to a smooth translation-type flow t · ( I, φ ) = (
I, φ + tξ ( I )) on the fibres of L . This local form, classicallyknown as action-angle coordinates, suggests that integrable flows are dynamicallyuninteresting. The example of the geodesic flow of a compact 3-dimensional Sol manifold which is completely integrable and has positive topological entropy, dueto Bolsinov and Taimanov [6], is proof that this is not the case. The presentpaper generalises the examples of [6, 11]. First, it shows how to construct inte-grable convex hamiltonian systems on cotangent bundles of certain solmanifoldsin higher dimensions that are analogues to the
Sol geometric 3-manifolds whenthe monodromy group is not R -split; second, it shows that Lax representationsof Bogoyavlenskij-Toda lattices are essential to construct these integrable systems,and moreover, the double-bracket Lax representations are essential to understandthe dynamics on the singular set; third, the Lax map of a Bogoyavlenskij-Todalattice and the ‘momentum map’ of a natural F -structure on the solmanifold forma dual pair; and, finally, the topological classification of these integrable systemscan be resolved by classifying abelian groups of Anosov toral automorphisms bythe topological entropy function.This appears to be a novel and interesting phenomenon: the construction of theseintegrable systems uses the machinery of Lax representations and R-matrices, whiletheir dynamical classification uses machinery developed to understand hyperbolicdynamical systems.Let us now sketch the constructions and results of the present paper.1.1. The
Sol -manifolds.
Let A be a torsion-free, abelian group of diffeomor-phisms of T b . The group A acts on T b × A R , where A R = A ⊗ Z R , via the diagonalaction ∀ α ∈ A, y ∈ T b , x ∈ A R : α ⋆ ( y, x ) := ( α ( y ) , x + α ⊗ . (1.1)This action is free and proper. The compact, smooth quotient is denoted by Σ orΣ A . The fibring of Σ by the tori T b equips Σ with a natural F -structure.Henceforth it is assumed that A <
GL( b ; Z ) is an abelian group of semi-simpleelements, hence contained in a Cartan subgroup of GL( b ; C ), and therefore an Date : October 29, 2018. exponential subgroup of GL( b ; C ). When A is an exponential subgroup of GL( b ; C ),the universal cover ˜Σ of Σ admits the structure of a solvable Lie group as follows:When A is an exponential subgroup of GL( b ; C ), A R is naturally identified with anabelian subgroup of GL( b ; R ). From this, there is a natural Lie group structure on R b ⋆ A R =: S and Z b ⋆ A =: S Z is a lattice subgroup of S , c.f. A contains a finite-index exponential subgroup A [26, Theorem 4.28]. Thefinite covering Σ of Σ induced by A has a universal cover with a solvable Liegroup structure; in this case, the fundamental group of Σ need not embed as asubgroup in this universal cover, although it does act as a free and proper groupof deck transformations [26, pp.s 70-71]. An elementary argument shows that if Γis a finite group of deck transformations and ϕ is an integrable flow on M that isΓ-invariant, then the induced flow on M/ Γ is integrable, also. So, to simplify thediscussion in this introduction, without losing generality, it will be assumed that A is an exponential subgroup of GL( b ; C ).1.2. Integrable geodesic flows.
Let y i be coordinates on C n which diagonalise A . Define complex-valued differential 1-forms on Σ by ν i = exp( − h ℓ i , x i ) d y i , and η i = d x i (1.2)where ℓ i ∈ Hom( A R ; R ) is the linear form which maps x ∈ A R to the logarithm ofthe modulus of its i -th eigenvalue and x i = h ℓ i , x i . A riemannian metric on Σ canbe defined by g = X i,j Q ij ν i · ν j + X i,j R ij η i · η j + X i,j S ij η i · ν j , (1.3)where Q, R and S are constant, complex symmetric matrices chosen so that g is areal, symmetric, positive-definite (0 , g is the general form ofa left-invariant metric on S = R b ⋆ A R . When the off-diagonal term S vanishes, thesubgroups R b and A R are orthogonal, totally geodesic and flat. By left-invarianceof g , each left translate of these two subgroups share these properties. Question A.
Which metrics g have a completely integrable geodesic flow? Some answers are known. The examples of Bolsinov and Taimanov shows thatwhen A is a cyclic group, then the geodesic flow is completely integrable for g with S = 0 and Q, R arbitrary [6, 7]. The present author showed that when A has rank b − A is R -split), Q ij = δ ij ǫ i , R is of a special form and S = 0, then thegeodesic flow is completely integrable. To explain the special form of R , write theHamiltonian of g in canonical coordinates:2 H g = X ij Q ij ∗ exp( h ℓ i + ℓ j , x i ) + X ij R ij p x i p x j , (1.4)where Q ij ∗ = Q ij p y i p y j (no sum). Because y i is a cyclic variable, p y i is a firstintegral. H g reduces to a family of Bogoyavlenskij-Toda-like Hamiltonians in thecanonical variables ( x, p x ). If one diagonalises Q , then the complete integrabilityof the Bogoyavlenskij-Toda Hamiltonian dictates the form of R . The introductionof [11] has an explicit example. The work of Adler & Van Moerebeke and Kozlov& Treschev suggests that when S = 0 the only completely integrable Hamiltonians H g arise from Bogoyavlenskij-Toda lattices or their deformations [3, 23, 22].The preceding argument glosses over a subtlety: the cyclic variables p y i aredefined only on the universal cover. In the above cases, one can construct smoothintegrals that descend to the quotient; this is true in general, but the difficulty liesin choosing R . This is related to Lax representations. This is also referred to as the Toda lattice or the Kostant-Toda lattice, but Kostant in [21]attributes to Bogoyavlenskij [4] the recognition of the role played by root systems of semisimpleLie algebras.
NTROPY AND TODA LATTICES 3
Lax Representations and momentum maps.
On the covering ˆΣ = T b × A R , there is the obvious free action of T b . This induces an F -structure on Σ, whichone may think of as a locally-defined free action of T b . The momentum map ˆ f ofthe T b -action induces a map f by equivariance, as illustrated in the right-hand sideof (1.5): L ∗ R (cid:15) (cid:15) T ∗ ˆΣ ˆΠ (cid:15) (cid:15) ˆ L o o ˆ f / / Lie( T b ) ∗ (mod A ) (cid:15) (cid:15) (mod ∼ ) ' ' OOOOOOOOOOO L ∗ R T ∗ Σ L o o f / / Lie( T b ) ∗ /A coll . / / Lie( T b ) ∗ / ∼ . (1.5)Lie( T b ) ∗ /A is neither a smooth manifold nor a Hausdorff space but it does containan open and dense subspace that is a smooth manifold. One can collapse thesingular set of Lie( T b ) ∗ /A to a single point to define a Hausdorff topological spaceLie( T b ) ∗ / ∼ , which is a smooth manifold outside of a single point, as illustrated infigure 1. Since the collapse is A -invariant, the map f is defined naturally. The map f is a first integral of H g and one can loosely think of f as the momentum map ofthe locally-defined T b action on T ∗ Σ.PSfrag replacements regular orbitssingular orbitsmod ∼ reduced space: Lie( T b ) ∗ / ∼ unreduced space: Lie( T b ) ∗ Figure 1.
The quotient map from Lie( T b ) ∗ → Lie( T b ) ∗ / ∼ .The regular points are points with a non-zero component in eacheigenspace of A ; the singular set is the complement.On the left of (1.5) is a map L , called a Lax matrix, that is implicit in theidentification of H g with a Bogoyavlenskij-Toda Hamiltonian. The construction ofa Poisson Lax map that Poisson commutes with f is the key difficulty in provingthe complete integrability of H g . Question B.
What conditions on A imply the existence of a Poisson Lax map L such that the T b -momentum map f and L form a dual pair? Implicit in the two papers of Bolsinov and Taimanov is the fact that if A is cyclic,then this question is trivially soluble. In [11, p. 529], the present author shows thatthere is a Poisson map that Poisson commutes with the T b -momentum map f when A <
GL( b ; Z ) is R -split and of finite index in its centraliser (the relation to Lax LEO T. BUTLER maps is hinted at in the remark on [11, p. 529]). To generalise that construction,it appears necessary to use the machinery of Lax representations.PSfrag replacements T b × T b × v geodesic v · y y Figure 2.
A returnmap.1.3.1.
Positive topological entropy.
The geodesic flow of g must have positive topological entropy, since π (Σ) has ex-ponential word growth [14]. When S = 0, there is a directproof of this: since A R is flat and totally geodesic in ˜Σ, asare all its left-translates, each curve t tv + y for v ∈ A R , y ∈ R b is a geodesic. On Σ, for v ∈ A , the geodesic isperiodic and one sees that the geodesic flow induces the re-turn map on T b defined by y v · y – which is a partiallyhyperbolic, and generally Anosov, automorphism of T b (seefigure 2).The appearance of such ‘subsystems’ heavily constrainsthe topological conjugacy class of a completely integrablegeodesic flow of the form of g .1.4. Results.
Let us sketch the main theorems of this paper.1.4.1.
Complete integrability.
Definition 1.
A torsion-free abelian subgroup
A <
GL( b ; Z ) is maximal if itselements are semisimple and it is of finite index in its centraliser. Theorem 1. If A <
GL( b ; Z ) is a maximal subgroup, then there is a Poisson Laxmap L such that (1.5) describes a dual pair, see (3.14). In particular, there is areversible Finsler metric on Σ A whose geodesic flow is completely integrable.If, in addition, an irreducible element in A has exactly r real eigenvalues and c non-real eigenvalues, then the geodesic flow of the riemannian metric g (1.3) iscompletely integrable when its Hamiltonian is defined as in (3.16) with root system g ( m ) in the cases described by Table 1.In all cases, the singular set is a real-analytic variety. Consequently, the inte-grable systems are semi-simple in the sense of [10] . r c g ( m ) r c g ( m ) ∗ A (1) n , D (2) n +1 D (3)4 ∗ A (1) n , D (2) n +1 ∗ B (1) n , C (1) n ∗ A (2)2 n ∗ A (2)2 n − ∗ D (1) n Table 1.
Conditions on eigenvalues and root systems which pro-duce a riemannian metric with integrable geodesic flow ( ∗ is anarbitrary positive integer).The first row in the upper left corner of the table summarises the result of [11].In the cases not covered in the table, it is uncertain if Σ A admits a riemannianmetric with completely integrable geodesic flow – the construction here yields onlycompletely integrable geodesic flows of reversible Finslers. If not, one would havethe first example of a compact smooth manifold that admits a completely integrablereversible Finsler, but not a riemannian, geodesic flow. NTROPY AND TODA LATTICES 5
Topological Entropy and Iso-energetic Topological Conjugacy.
The tangentspaces to the T b -fibres of Σ form a sub-bundle V ⊂ T Σ. Let V ⊥ ⊂ T ∗ Σ be theannihilator of V . The subspace V ⊥ is invariant under the geodesic flow of Theorem1 and that geodesic flow has positive topological entropy on V ⊥ . We prove Theorem 2.
Let
Σ = Σ A be as in Theorem 1 and let ϕ be the Hamiltonian flowon T ∗ Σ induced by the Hamiltonian H defined in (3.16). Then (1) V ⊥ is a weakly normally hyperbolic invariant manifold. Its stable and un-stable manifolds coincide and equal the pre-image of the equivalence classof in Lie( T b ) ∗ / ∼ under the T b -momentum map f ; (2) the topological entropy of ϕ | H − ( ) equals that of ϕ | V ⊥ ∩ H − ( ) , when H is induced by the A (1) n Bogoyavlenskij-Toda lattice; (3) in all cases, the topological entropy of ϕ | V ⊥ ∩ H − ( ) is calculable (seeTable 5). The construction of the Lax map in theorem 1 is unique up to the action of apermutation group. If φ , φ are two such permutations and ϕ , ϕ are the result-ing geodesic flows, an interesting question is whether these flows are topologicallydistinct. A topological invariant, namely the marked homology spectrum, doesdistinguish these flows in many cases even when topological entropy cannot. Toexplain, let h( v ) = h top ( v ) h : A → R (1.6)be the entropy function, where v ∈ A is viewed as a T b -automorphism. Theorem 3.
If there is a topological conjugacy of ϕ , ϕ on their respective unitsphere bundles, then there is an automorphism f : A → A such that h ◦ f = h . (1.7) If the number-theoretic closure of A is a group of Anosov automorphisms, then f isinduced by a Galois automorphism. In many cases, the group of Galois automorphisms is trivial, which implies thateach of the constructed Hamiltonian flows must be topologically non-conjugate. Ingeneral, one should not expect the number-theoretic closure of A to be a group ofAnosov automorphisms, though. Question C.
Which automorphisms of A fix the entropy function h ? This leads to a further question, whose formulation is somewhat technical andis deferred to section 7, question F. Finally, theorem 7.3 provides information onthe number of distinct topological conjugacy classes of integrable Hamiltonian flowsprovided by theorem 1.Question C is a rigidity question: to what extent does the entropy of an actiondetermine that action. An approach to this question is to ask which embeddingsof Z a ∼ = A into GL( b ; Z ) have equal entropies. Katok, Katok and Schmidt giveexamples of iso-entropic actions of Z on T by maximal subgroups of GL(3; Z )that are conjugate in GL(3; Q ) but not conjugate in GL(3; Z ) [20]. However, thesuspension manifolds of these actions are not homotopy equivalent. Indeed, if A ′ < GL( b ; Z ) is not conjugate to A in GL( b ; Z ), then π (Σ A ′ ) is not isomorphic to π (Σ A ). Thus, question C is somewhat finer than the iso-entropic rigidity problemexamined in [20]. LEO T. BUTLER
Two additional questions.
Let Σ be a torus bundle over a torus of thetype described in section 1.1. Σ is aspherical and the fundamental group of Σ isa poly- Z group , so Theorem 15B.1 of [29] implies that π (Σ) determines Σ upto homeomorphism. The standard smooth structure on Σ is defined by the aboveconstruction. In general, the topological manifold Σ may admit several inequivalentsmooth structures. Question D.
Which smooth structures on the topological manifold Σ admit a rie-mannian or Finsler metric whose geodesic flow is completely integrable? This question is already quite interesting when A = 1 and Σ is a torus since[12] shows that the integrals cannot all be real-analytic if the smooth structure isnon-standard. It is unknown if there are analogous obstructions when A = 1.And, finally, Question E.
What conditions on
A <
GL( b ; Z ) imply that Σ A admits a riemann-ian or Finsler metric whose geodesic flow is completely integrable? Theorem 4.1 shows that there are natural examples of groups A that are notmaximal, yet Σ A admits a completely integrable Finsler. These examples are con-structed using symmetries provided by number-theoretic considerations. The diffi-culty in the general case, where there are no obvious symmetries, is the constructionof the Lax map L appears to break down.2. Notation and Preliminary Definitions
Integrability.
The present paper’s definition of complete integrability followsthat of [5, 11].Let Σ be a real-analytic manifold. The set of smooth functions on the cotangentbundle of Σ, C ∞ ( T ∗ Σ), has two canonical algebraic structures: it is an abelianalgebra when equipped with the natural operations of point-wise addition and mul-tiplication; and, coupled with the canonical
Poisson bracket , { , } , ( C ∞ ( T ∗ Σ) , { , } ) isa Lie algebra of derivations of the algebra C ∞ ( T ∗ Σ). A hamiltonian H ∈ C ∞ ( T ∗ Σ)induces a vector field Y H := { , H } . For A ⊂ C ∞ ( T ∗ Σ) and P ∈ T ∗ Σ, let d A P = span { df P : f ∈ A} and let Z ( A ) = { f ∈ A : {A , f } ≡ } . Let k = sup P dim d A P , l = sup P dim dZ ( A ) P . Let us say P ∈ T ∗ Σ is A - regular if thereexist f , . . . , f k ∈ A such that P is a regular value for the map F = ( f , . . . , f k )and f , . . . , f l ∈ Z ( A ); if P is not A -regular then it is A - critical . Let L ( A ) be theset of A -regular points. H is assumed to be proper. Definition 2 ( c.f. [5]) . H ∈ C ∞ ( T ∗ Σ) is integrable if there is a Lie subalgebra A ⊂ C ∞ ( T ∗ Σ) such that: (1) H ∈ Z ( A ) ; (2) k + l = dim T ∗ Σ and L ( A ) is an open and dense subset of T ∗ Σ .If k = l = dim Σ , we will say that H is completely integrable . Bolsinov and Jovanovic [5] introduced this definition of complete integrability.The standard definition of complete integrability (resp. non-commutative integra-bility) are special cases of Definition 2 with A = span { f , . . . , f k } and l = k (resp. l ≤ k ) and the regular-point set of F = ( f , . . . , f k ) is dense. Definition 2 is bothmore intrinsic, and more suited to the examples of the present paper. Note that thepresent definition of integrability is equivalent to that of Dazord & Delzant [13]. That is, there is a sequence of subgroups 0 = D m ⊳ D m − ⊳ · · · ⊳ D = D such that D i / D i +1 ∼ = Z for all i . NTROPY AND TODA LATTICES 7
Construction of the solmanifolds and number theory.
There is a well-known correspondence between abelian subgroups of GL( b ; Z ) and groups of unitsin algebraic number fields of degree d dividing b [20]. The present paper exploitsthis correspondence extensively. The following section establishes notation thatis used throughout. In terms of the terminology in the introduction, we use thefollowing translation table:abelian A <
GL( b ; Z ) → a group of units in the field generated by theeigenvalues of all a ∈ A ; Z b → a direct sum of copies of a subgroup of theintegers of a number field.2.2.1. Preliminaries.
Let Q ⊂ F ι ⊂ E be an inclusion of algebraic number fields. For a field extension E/F let the set ofembeddings of E into C which fix F be denoted by G E/F ; we adopt the convention / Q is omitted. Define vector spacesW E = X σ ∈ G E C σ, (2.1)and V E = { x ∈ W E : x ¯ σ = ¯ x σ ∀ σ ∈ G E } , (2.2)where ¯ denotes complex conjugation and ¯ σ is the embedding σ followed by complexconjugation. We also defineV o,E = { x ∈ V E : X σ ∈ G E x σ = 0 , & x σ = x ¯ σ ∀ σ ∈ G E } . (2.3) G E is a basis of V E which induces the dual basis G ∗ E of V ∗ E . An element in thedual basis shall be denoted by ˆ σ for σ ∈ G E . The basis and dual basis establish alinear isomorphism between V E and V ∗ E which shall be denoted by the circumflexoperator, V E → V ∗ E : x ˆ x , whose inverse is V ∗ E → V E : x ˇ x .One obtains a basis of V ∗ o,E as follows: note that ˆt = | G E | P σ ∈ G E ˆ σ and ˆ σ − ˆ¯ σ vanish on V o,E for all σ ∈ G E . If one defines G rE to be the set of real embeddingsof E and G cE to be one-half of the non-real embeddings such that G cE is disjointfrom its complex conjugate, then one observes thatV ⊥ o,E = R · ˆt ⊕ X σ ∈ G cE R · (ˆ σ − ˆ¯ σ ) , V ∗ o,E = X σ ∈ B F R · ˆ σ | V o,E (2.4)where B E = G rE ∪ G cE .The inclusion F ι ⊂ E inducesV E ι ∗ / / / / V F where ι ∗ ( σ ) = σ | F , and V ∗ E o o ι ? _ V ∗ F where ι (ˆ τ ) = X σ ∈ G E ,σ | F = τ ˆ σ. (2.5)Finally, define a map V ∗ E α / / / / V ∗ o,F by α = ∗ ˆ ι ∗ , ˆ σ ˆ τ | V o,F where τ = σ | F , (2.6) LEO T. BUTLER ∗ is the adjoint of the inclusion map V o,F ⊂ V F and ˆ ι ∗ = ˆ ι ∗ ˇ. This allows one todefine a pairing between V ∗ E and V o,F , denoted as follows h ˆ σ, x i := h α (ˆ σ ) , x i ∀ σ ∈ G E , x ∈ V o,F , = h ˆ τ , x i where τ = σ | F . (2.7)Since x = P τ ∈ G F x τ · τ , it is apparent that h ˆ σ, x i = x ( σ | F ) ∀ σ ∈ G E , x ∈ V o,F , (2.8)so the notation is natural.2.3. An embedding of O E in V E . Let O E be the ring of integers of E , and let U E be the group of multiplicative units of O E . Define a map η : O E → V E by η ( α ) := X σ ∈ G E σ ( α ) · σ, (2.9)for each α ∈ O E . Lemma 2.1.
The map η is an embedding whose image—call it N E —is a discrete,cocompact subgroup of V E .Proof. This is standard. (cid:3)
Let T E = V E / N E be the resulting torus. T E is equipped with a canonical affinestructure from V E and the group U E acts by automorphisms of T E defined by u · y = X σ ∈ G E σ ( u ) · y σ · σ + N E , (2.10)where y = P σ ∈ G E y σ · σ + N E is an element in T E and u ∈ U E . The actionin (2.10) is well-defined since N E is mapped to itself by U E . A fortiori , equation(2.10) also defines an action of U F ⊂ U E as an abelian group of automorphisms of T E .2.4. An embedding of U + F in V o,F . Define a map ℓ : U F → V F by ℓ ( u ) = X σ ∈ G F ln | σ ( u ) | · σ. (2.11)Since ¯ σ is σ followed by complex conjugation, it is clear that LLL F =: im ℓ ⊂ V o,F .Dirichlet’s theorem on the group of units of an algebraic number field characterisesthe image of ℓ as a discrete, cocompact subgroup of V o,F , while ker ℓ =: R F is theset of units all of whose conjugates lie on the unit circle. Stated otherwise, there isan unnatural splitting of U F via a commutative diagram R F (cid:31) (cid:127) / / = (cid:15) (cid:15) U F / / / / = (cid:15) (cid:15) U F / R F ℓ ∼ = / / ∼ = (cid:15) (cid:15) LLL F , ∼ = (cid:15) (cid:15) R F (cid:31) (cid:127) / / R F ⊕ U + F / / / / U + F ∼ = / / Z r + c − , where r (resp. 2 c ) is the number of real (resp. non-real) embeddings of F . When F has a real embedding, which one may take to be the identity embedding F ⊂ C ,then R F = {± } and U + F may be taken to be the multiplicative group of positiveunits in U F — hence the notation. To summarise Lemma 2.2.
The image of the map ℓ : U + F → V o,F — call it LLL F — is a discrete,cocompact subgroup of V o,F isomorphic to U + F . NTROPY AND TODA LATTICES 9
An action of U + F on T E × V o,F . For y ∈ T E , x ∈ V o,F and u ∈ U + F define u · ( y , x ) := ( u · y , x + ℓ ( u )) . (2.12)This action is clearly free and proper. Let Σ denote the compact manifold obtainedby quotienting T E × V o,F by this action of U + F . Lemma 2.3.
There is a commutative diagram of natural maps V E × V o,F / / / / (cid:127) _ = (cid:15) (cid:15) (cid:15) (cid:15) T E × V o,F / / / / (cid:127) _ = (cid:15) (cid:15) (cid:15) (cid:15) ( T E × V o,F ) / U + F (cid:127) _ = (cid:15) (cid:15) (cid:15) (cid:15) ˜Σ ˜ π / / / / ˆΣ ˆ π / / / / Σ . (2.13) Therefore, π (Σ) is naturally isomorphic to the semi-direct product ∆ = U + F ⋆ O E ,while there is a natural fibring of Σ by tori over a torus T E (cid:31) (cid:127) / / Σ p / / / / T o,F , (2.14) where T o,F = V o,F / LLL F .Proof. Naturality of the construction implies the lemma. (cid:3)
The cotangent bundle T ∗ Σ . The vector space structures on V E and V o,F give a tautological trivialisation of their cotangent bundles. Lemma 2.3 thereforeimplies that there is a commutative diagramV ∗ E × V E × V ∗ o,F × V o,F / / / / (cid:127) _ = (cid:15) (cid:15) (cid:15) (cid:15) V ∗ E × T E × V ∗ o,F × V o,F / / / / (cid:127) _ = (cid:15) (cid:15) (cid:15) (cid:15) (V ∗ E × T E × V ∗ o,F × V o,F ) / U + F (cid:127) _ = (cid:15) (cid:15) (cid:15) (cid:15) T ∗ ˜Σ ˜Π / / / / T ∗ ˆΣ ˜Π / / / / T ∗ Σ , (2.15)where ˆΠ is the covering map induced by ˆ π , etc. Let us introduce coordinates on T ∗ ˆΣ by P ∈ T ∗ ˆΣ ⇐⇒ P = ( Y , y + N E , X , x ) ∈ V ∗ E × T E × V ∗ o,F × V o,F . The action of U + F on T ∗ ˆΣ is the natural lift of the action on Σ u · P = ( u · Y , u · y + N E , X , x + ℓ ( u )) (2.16)where u · y is defined in equation (2.10) and u · Y = P σ ∈ G E Y σ · σ ( u ) − · ˆ σ is theinduced contragredient action.2.7. Functions on T ∗ Σ . The function P X is U + F -invariant, so one may view X as a submersion T ∗ Σ ։ V ∗ o,F .Fix a positive integer b σ for each σ ∈ G E and define the function γ σ ( P ) := exp( b σ · h ˆ σ, x i ) × | Y σ | b σ (2.17)where the pairing h ˆ σ, x i is defined in equation (2.7). Lemma 2.4.
The function γ σ is U F -invariant and it is real-analytic if b σ is even.Proof. From equation (2.16), we know that for each u ∈ U F γ σ ( u · P ) = γ σ ( P ) × exp( b σ ln | σ ( u ) | ) × | σ ( u ) | − b σ = γ σ ( P ) . (2.18)It is clear that exp( b σ · h ˆ σ, x i ) is real-analytic, and | Y σ | b σ is real-analytic if b σ is apositive even integer. (cid:3) Remark 2.1.
Fix even integers b σ as in lemma 2.4. One may define a momentum-like map λ : T ∗ Σ → V ∗ o,F ⊕ V ∗ E by λ ( P ) = X ⊕ X σ ∈ G E γ σ ( P ) · ˆ σ, ∀ P ∈ T ∗ Σ . (2.19)When the universal cover ˜Σ of Σ admits the structure of a solvable Lie group with∆ as a lattice subgroup, V ∗ o,F ⊕ V ∗ E – as the dual of a Lie algebra – admits acanonical Poisson structure. In this case, the map λ is left-invariant and Poissonand therefore mimics the properties of the classical momentum map.3. Lax Representations
Real split affine Lie algebras.
Let us briefly recall the construction underly-ing the Lax representation of periodic Bogoyavlenskij-Toda lattices. This discussionfollows that in [27, 1, 2]. Let g be a simple real Lie algebra with the real split Cartansub-algebra h ; g is also known as the real normal form of the simple complex Liealgebra g ⊗ C . The Cartan-Killing form of g is denoted by hh , ii when viewed asa bilinear form on g , and it is denoted by κ when viewed as a linear isomorphismof g with g ∗ . Recall that hh , ii is non-degenerate on h . As h is a real split Cartansub-algebra, g decomposes as g = h + X r ∈ Ψ ∗ g r (3.1)where Ψ ∗ ⊂ h ∗ is the set of roots and g r is the root space associated with r , g r = { x ∈ g : ad h x = h r, h i x ∀ h ∈ h } . There is a set of simple roots Ψ ⊂ Ψ ∗ such that every root is an integer linear combination of the roots in Ψ with entirelynon-negative or non-positive coefficients. The height of a root is the sum of thesecoefficients; there is a unique root, η , of minimal height. Let Ψ be Ψ ∪ { η } . Define L to be the set of Laurent polynomials in the variable λ with coefficientsin g ; L inherits an obvious Lie algebra structure. Let d = λ ∂∂λ be a derivation;define [d , x · λ n ] = nx · λ n for all integers n and x ∈ g . Then ˆ g = L + R · d is a realsplit Lie algebra with Cartan sub-algebra ˆ h = h + R · d. The Cartan sub-algebrainduces a weight-space decomposition of L as L = h + X r ∈ ΨΨΨ ∗ L r (3.2)where ΨΨΨ ∗ = n r ∈ ˆ h ∗ : r | h ∈ Ψ ∗ ∪ { } , h r , d i ∈ Z , r = 0 o . The weight set ΨΨΨ ∗ has abasis of simple weights ΨΨΨ = Ψ ∪ { ηηη } , where ηηη | h = η and h ηηη, d i = 1. Each r ∈ ΨΨΨ ∗ isan integer linear combination of roots in ΨΨΨ. By defining the height of r as the sumof these coefficients one obtains the principal grading L = X n ∈ Z L n , (3.3)where L = h , L n = P ht( r )= n L r otherwise, and [ L n , L m ] ⊆ L m + n for all m, n . Itis observed that L ± = g η λ ± + X r ∈ Ψ g ± r = X r ∈ ΨΨΨ R · e ± r (3.4)the same sign appearing throughout, and e r is a vector normalised so that κ · [e r , e − r ] ∈ Ψ . The sub-algebras L + = P n ≥ L n , L − = P n< L n permit thedefinition of a second Lie algebra structure on L , defined by[ x, y ] R := [ x + , y + ] − [ x − , y − ] (3.5)for x = x + + x + , y = y − + y + ∈ L − ⊕ L + . The Cartan-Killing form κ allows oneto identify L ∗ n = L − n for all n , in such a way that κ (e r ) = e − r or hh e r , e − s ii = δ r , s . NTROPY AND TODA LATTICES 11
Indeed, note that e r = e r · λ n where r = r | h and n = h r , d i for all roots r ∈ ΨΨΨ, soit suffices to find a suitable basis of g in order to define the vectors e r . One alsoknows that Lemma 3.1.
For each µ ∈ L ∗− , the affine subspace µ + L ∗ + L ∗ is a Poissonsubspace of L ∗ R . The Casimirs of L ∗ are in involution on L ∗ R .Proof. See references [27, 1]. (cid:3)
A second splitting.
Let L = h + P r ∈ ΨΨΨ ∗ R · e r , a sub-algebra of the loopalgebra L on which the Cartan-Killing form is non-degenerate. One can distinguishtwo sub-algebras L ± such that L = L − ⊕ L + as a vector space: L − = P r ∈ ΨΨΨ + R · (e r − e − r ) , L + = h + P r ∈ ΨΨΨ + R · e r , so L ∗− ≡ L ⊥ + = P r ∈ ΨΨΨ + R · e r , L ∗ + ≡ L ⊥− = h + P r ∈ ΨΨΨ + R · (e r + e − r ) , (3.6)where ΨΨΨ + ⊂ ΨΨΨ ∗ is the set of positive roots. One can define a grading on both L ± by defining the height of a root r ∈ ΨΨΨ + to be ht( r ) = ht( r ) + (1 + k ) h r , d i where k is the height of the maximal root of g . With this grading, a basis of L +1 (resp. L − ) is { e r : r ∈ ΨΨΨ } (resp. { e r − e − r : r ∈ ΨΨΨ } ) while a basis of L ∗ +1 (resp. L ∗− )is { e r + e − r : r ∈ ΨΨΨ } (resp. { e − r : r ∈ ΨΨΨ } ). One therefore knows that L admitsan R -bracket analogous to that defined in (3.5) and that Lemma 3.1 also holds for L R . Remark 3.1. If α is an automorphism of the graded Lie algebra L that fixes h , thenthe fixed point set of α is a sub-algebra that inherits a grading, splitting and a rootspace decomposition from L . The constructions of both subsections 3.1 and 3.2 areapplicable in this case, too. The automorphism α satisfies α ( x · λ n ) = α ( x ) · ( ǫλ ) n for all x ∈ g and n , where ǫ is a primitive order( α ) root of unity. This constructionyields the so-called twisted loop algebras. The twisted loop algebra is traditionallydenoted by g ( m ) where m is the order of the automorphism α ; when m = 1, onehas the usual loop algebra L .3.3. Examples.
Let g = A = sl (3; R ). For h one can take the sub-algebra oftrace zero diagonal matrices and for the basis of positive roots of g one can takethe roots r and r with the minimal root η : r = − , r = − , η = − r = − , (3.7)which satisfy the linear relation r + r + η = 0. A root r ∈ ΨΨΨ may be writtenformally as r = ± r i + n where n = h r , d i . The height of r is then computed to be3 n ± i = 1 , n ± i = 3. From this, one can see that the gradedpieces of L , as in (3.3), are L = h and L +1 = λa α λa α λα λa , L +2 = λ a α λα λ a λα λ a , L − = λ − a λ − α α λ − a α λ − a , L − = λ − a λ − α λ − a λ − α α λ − a , (3.8)where a i , α i are real numbers and P a i = 0. The splitting in (3.6) of L implies that the root spaces of height ± L − = α − α λ − − α α α λ − α , L ∗− = α
00 0 α α λ , L +1 = α
00 0 α α λ , L ∗ +1 = α α λ − α α α λ α , (3.9)where the α i are real.3.4. Bijections.
Assume that F/ Q is an algebraic field of degree m with r realembeddings and 2 c non-real embeddings such that r + c = n , and that g is a realsplit affine Lie algebra of rank n −
1. Since B F , the set of real embeddings of F plus one-half the set of complex embeddings of F , has n elements and the simpleroots of L , ΨΨΨ, have n elements, the sets are isomorphic. Definition 3.
Let B be the set of bijections B F → ΨΨΨ . Each ρ ∈ B can be extended to a map G F → ΨΨΨ by ρ (¯ σ ) := ρ ( σ ) for all σ ∈ B F .This extension shall be understood throughout.Additionally, each ρ ∈ B naturally induces a linear isomorphism φ = φ ρ : V ∗ o,F → h ∗ . To define φ let us recall two things. First, note that the projection ˆ h ∗ ։ h ∗ that is dual to the inclusion h ֒ → ˆ h induces the bijection ΨΨΨ ∼ = Ψ : r r = r | h .Second, there are unique positive integers ω r such that X r ∈ Ψ ω r r = 0 , gcd( ω r : r ∈ Ψ) = 1 . (3.10)For each τ ∈ B F , define n τ to be 1 if τ is a real embedding; and 2 if not. Then,define φ ( ˆ τ | V o,F ) = n − τ ω r r where r = ρ ( τ ) . (3.11)Since ˆ τ equals ˆ¯ τ when restricted to V o,F , the sole linear dependence relationamongst the set n ˆ τ | V o,F : τ ∈ B F o is the relation X τ ∈ B F n τ ˆ τ | V o,F = X τ ∈ G F ˆ τ | V o,F = 0 . Thus, equation (3.10) implies that φ extends to a linear isomorphism.3.5. Lax representations.
Fix ρ ∈ B and let φ = φ ρ be the induced linearisomorphism. Let Φ : V ∗ o,F → h ∗ be a linear map and let g ± : V ∗ E × G E → L ∗± besmooth maps. Define a map L = L ρ, Φ : T ∗ ˜Σ → L ∗ by κ − · L ( P ) = X σ ∈ G E g − ,σ ( Y ) · e r + Φ( X ) + X σ ∈ G E g + ,σ ( Y ) · exp( b σ ·h ˆ σ, x i ) · e − r (3.12)where it is understood that r = ρ ( σ | F ) in the sums and L ∗ is identified with L viathe Cartan-Killing form κ .There are several choices of Lax representation that are useful. The first is (inall cases, r = ρ ( σ | F ) is understood) κ − · L ( P ) = X σ ∈ G E | Y σ | b σ · e r + Φ( X ) + 12 × X σ ∈ G E exp( b σ · h ˆ σ, x i ) · e − r , (3.13)while the second is κ − · L ( P ) = X σ ∈ G E e r + Φ( X ) + 12 × X σ ∈ G E | Y σ | b σ · exp( b σ · h ˆ σ, x i ) · e − r , (3.14) NTROPY AND TODA LATTICES 13 and a third is κ − · L ( P ) = Φ( X ) + 1 √ × X σ ∈ G E | Y σ | b σ · exp( 12 b σ · h ˆ σ, x i )(e r + e − r ) . (3.15)Note that the Lax representations in (3.13–3.14) are related to the splitting of theloop algebra in section 3.1; the final Lax representation in (3.15) is related to thesplitting in section 3.2. In all cases, the pullback of the Casimir x × κ ( x, x ) on L ∗ by any of the three Lax matrices in (3.13–3.15) L is equal to H := 12 × hQ · X , X i + 12 X σ ∈ G E | Y σ | b σ · exp( b σ · h ˆ σ, x i ) , (3.16)where Q : V ∗ o,F → V o,F is defined by Q = Φ ∗ κ Φ. This Hamiltonian is fibre-wisequadratic — hence, induced by a riemannian metric — iff b σ = 2 for all σ ; in allcases, it is fibre-wise convex. The next theorem implies that there are constraintson F if H is fibre-wise quadratic.As a second step, recall that sl R has a basis h, e + , e − such that [ h, e ± ] = ± e ± ,and [e + , e − ] = 2 h . The Cartan-Killing form identifies the dual basis as h, e − , e + .For each σ ∈ G E , let sl R σ be a copy of sl R and let h σ , e ± ,σ be copies of h, e ± .Define L : T ∗ ˜Σ → g ∗ , g = X σ ∈ G E sl R σ by L ( P ) = X σ ∈ G E Y σ · h σ + exp( h ˆ σ, y i ) · e − ,σ . (3.17) Theorem 3.2. L is a Poisson map. L = L ρ, Φ : T ∗ ˜Σ → L ∗ R is a Poisson map iffthere is a c ∈ Z + such that: (1) for all σ ∈ G E and r ∈ ΨΨΨ with ρ ( σ | F ) = r , one has n − σ | F ) b σ ω r = c ; and (2) Φ = c − × φ ρ .The map L = L + L : T ∗ ˜Σ → L ∗ + g ∗ is a Poisson embedding if either g + or g − is an embedding and E = F .Proof. The proof shall assume that L is defined by equation (3.13); the remainingcases are not significantly different. To prove that L is a Poisson map, one needsto prove that { f ◦ L , g ◦ L } T ∗ ˜Σ = { f, g } h ∗ ◦ L , (3.18)for all smooth functions f, g on h ∗ . It suffices to verify equation (3.18) holds forlinear functions f, g , for a single copy of sl R , and a single pair of conjugate variables Y and y . For f = h and g = e + one sees that { h, e + } sl R ∗ ◦ L = − hh L , [ h, e + ] ii = − e y , (3.19)while { h ◦ L , e + ◦ L } T ∗ ˜Σ = { Y , e y } = − e y . (3.20)Since h and e + are functionally independent at almost all points on almost allco-adjoint orbits, this proves that L is a Poisson map.To prove the claim concerning L , one needs to prove that { f ◦ L , g ◦ L } T ∗ ˜Σ = { f, g } L ∗ R ◦ L , (3.21)for all f, g ∈ C ∞ ( L ∗ ). As above, it suffices to verify equation (3.21) holds for all f, g ∈ L R . Given the bracket relations on L R , it suffices to prove the equation forall f, g ∈ L − + L + L +1 . Let us break this into cases: (1) If f, g ∈ L or f ∈ L − and g ∈ L +1 or f ∈ L and g ∈ L − , then [ f, g ] R = 0so { f, g } L ∗ R ◦ L = 0. On the other hand, { f ◦ L , g ◦ L } T ∗ ˜Σ = 0 since functionsof X alone, or a function of Y and a function of x , or a function of Y and afunction of X alone Poisson commute.(2) If f, g ∈ L − or f, g ∈ L +1 , then [ f, g ] R ∈ L ± . Therefore, { f, g } L ∗ R ◦ L = − hh L , [ f, g ] R ii = 0 (3.22)since L lies in L − + L + L +1 . On the other hand, f ◦ L and g ◦ L are eitherfunctions of Y or x alone. In either case, they Poisson commute on T ∗ ˜Σ.(3) If f ∈ L and g ∈ L +1 , then it suffices to assume that g = e r for some r ∈ ΨΨΨ. In this case, { f, g } L ∗ R ◦ L = − h L , [ f, e r ] R i = − h r , f i × hh L , e r ii (3.23)= − h r, f i × X σ ∈ G E s . t . ρ ( σ | F )= r exp( b σ h ˆ σ, x i ) . On the other hand, f ◦ L = h X , Φ ∗ f i , (3.24) g ◦ L = X σ ∈ G E s . t . ρ ( σ | F )= r exp( b σ h ˆ σ, x i ) , (3.25)so the Poisson bracket of these functions is { f ◦ L , g ◦ L } T ∗ ˜Σ = − X σ ∈ G E s . t . ρ ( σ | F )= r exp( b σ h ˆ σ, x i ) × b σ h ˆ σ, Φ ∗ f i (3.26)Because ρ is a bijection of B F with ΨΨΨ, there is a unique τ ∈ B F suchthat ρ ( τ ) = r . Therefore, due to the way that ρ is extended to G F , the σ ’s involved in the above summations all satisfy σ | F = τ or ¯ τ . Therefore, h ˆ σ, x i = h ˆ τ , x i for all σ ∈ G E such that ρ ( σ | F ) = r .This fact about the σ ’s also implies that h ˆ σ, Φ ∗ f i = (cid:10) Φ(ˆ τ | V o,F ) , f (cid:11) . Sinceˆ τ will only appear when it acts on V o,F , the notation | V o,F will be sup-pressed.Therefore, if the right-hand sides of equations (3.23) and (3.26) areequated for all f ∈ L , then one concludes that X σ ∈ G E s . t . ρ ( σ | F )= r exp( b σ h ˆ τ , x i ) × [ b σ Φ(ˆ τ ) − r ] = 0 . (3.27)The functions u e au , e bu are linearly independent if a = b . If the b σ ’s inthe above sum are not constant, then the sum in equation (3.27) containstwo linearly independent functions. Therefore, the coefficients on these twofunctions must vanish. But this forces Φ(ˆ τ ) to equal two different multiplesof r . Absurd. Therefore, the b σ ’s in the sum must be constant. This impliesthat b σ is determined by r alone, or equivalently, by τ alone.As cases (1—3) are the only independent cases to be considered, one concludesthat if L is a Poisson map, then there are integers b τ , τ ∈ B F , such that the integers b σ , σ ∈ G E , satisfy b σ = b τ where σ | F = τ or ¯ τ . (3.28)Moreover, equation (3.27) implies that if τ ∈ B F and ρ ( τ ) = r , thenΦ(ˆ τ ) = b − τ r. (3.29) NTROPY AND TODA LATTICES 15
Summing over τ ∈ G F , and using the fact that ρ is a bijection of B F and ΨΨΨ, X τ ∈ G F Φ(ˆ τ ) = X τ ∈ B F Φ( n τ ˆ τ ) = X r ∈ Ψ n τ b − τ r, (where ρ ( τ ) = r ) . (3.30)The left-hand side vanishes because P τ ∈ G F ˆ τ | V o,F = 0. Therefore X r ∈ Ψ n τ b − τ r = 0 , (3.31)while the unique linear dependence relation in equation (3.10) implies that theremust be a constant c such that n − τ b τ ω r = c for all r ∈ Ψ. The constant c is apositive integer, or one-half such, since b τ and ω r are positive integers and n τ = 1or 2. This implies part (1) of the Theorem.The equation that Φ must satisfy is, for all τ ∈ B F ,Φ(ˆ τ ) = 1 cn τ × ω r r where r = ρ ( τ ) . (3.32)Comparison with equation (3.11) shows that Φ = c − × φ ρ , which is part (2) of theTheorem.The claim that L = L + L is an embedding is obvious. (cid:3) Remark 3.2.
Theorem (3.2) exploits the naturality of the constructions. In caseswhere the group A is not of finite index in U F , one encounters the problem thatthere is no obvious Lax map. This is what makes question D difficult. Remark 3.3.
Condition (1) implies that b σ depends only on σ | F . Condition (2)implies that 2 c is divisible by all ω r , hence by their lcm, ω . Therefore, there is aunique choice of Φ and integers b σ if one insists that c be as small as possible andthe b σ be even. In case F is totally real, condition (1) implies that c is divisibleby ω = lcm( ω r : r ∈ Ψ). When c is chosen to be 2 ω – so that b σ = 2 ω/ω r is even–, condition (2) implies that Φ( ˆ σ | F ) = w r r , where w r = ω r /ω , and ρ ( σ | F ) = r .This condition is stated in [11, Lemma 7], except for the factor of in Φ. Thisdiscrepancy is due to the choice of a slightly different Poisson structure in [11,equation (9-10)]. With these choices, the Hamiltonian H in equation (3.16) isequal to that in [11, equation 15] when E = F is totally real. In particular ( c.f. equation 3.16), Q = φ ∗ ρ · κ · φ ρ . (3.33)In the event that F has no real embeddings, this smallest choice is c = ω and b σ = 2 ω/ω r ; in other events, the solution is somewhat more involved to state, asit depends on the bijection ρ (see tables 2–4 in section 4.1.3 for examples). It ispossible to state Proposition 3.1.
Let r be the number of real embeddings of F , and c the numberof non-real embeddings. If the Hamiltonian H is fibre-wise quadratic, then table (1) is true. In particular, if r > and c > , then none of the Hamiltonians H arefibre-wise quadratic.Proof. From equation (3.16), it is clear that H is fibre-wise quadratic iff b σ = 2 forall σ . Since, for all roots r , ω r = n τ c/ τ = σ | F and ρ ( τ ) = r , one has τ ∈ G cF : τ ∈ G rF : ω r = c ω r = c/ . The coefficients b σ in [11] are one-half those in the present paper. This shows that each weight is 1 or 2. If c = 0, then c = 2 and ω r = 1 for all roots.If r = 0, then c = 1 and ω r = 1 for all roots. If r , c >
0, then c = 2 and ΨΨΨ has r roots with weight 1 and c roots with weight 2. Inspection of the root systems infigures (8–9) completes the proof. (cid:3) Quotients of the Lax Representations.Lemma 3.3.
There is a natural action of ∆ = U + F ⋆ O F —which factors through U + F —on L ∗ R such that the map defined in equation (3.13) is ∆ -equivariant, henceinduces a Poisson map T ∗ ˜Σ L / / ˜Π (cid:15) (cid:15) L ∗ R (cid:15) (cid:15) T ∗ Σ L / / L ∗ R / U + F . The action of U + F on im L ⊂ L ∗ R is free and proper.Proof. Define the action of g = ( u, α ) ∈ U + F ⋆ O F on L ∗ R by g · e r = | σ ( u ) | − b σ · e r if r ∈ ΨΨΨ , ρ ( σ ) = r | σ ( u ) | b σ · e r if − r ∈ ΨΨΨ , ρ ( σ ) = r e r otherwise , (3.34)and g | L = 1. It is straightforward to see that L ( g · P ) = g · L ( P ) for all g and P ∈ T ∗ ˜Σ.Since the coefficients of e r , − r ∈ ΨΨΨ, do not vanish on im L , one sees that theaction of U + F is semi-conjugate to its action on V o,F . Hence, it is free and proper. (cid:3) Remark 3.4.
The preceding lemma implies that H and all the “spectral” integralsof H descend, but with some additional work. The alternative Lax matrix, equation(3.14), gives us a simple proof of this fact. Lemma 3.4.
The map defined in equation (3.14) is U + F ⋆ O F -invariant, hence itinduces a Poisson map T ∗ ˜Σ L / / ˜Π (cid:15) (cid:15) L ∗ R T ∗ Σ . L ooooooooooooo Consequently, if h is a Casimir of L ∗ , then h ◦ L Poisson commutes with H .Proof. By equation (2.17), one can write L ( P ) = X σ ∈ G E e r + Φ( X ) + 12 × X σ ∈ G E γ σ · e − r . (3.35)By Lemma 2.4, each function γ σ is U + F -invariant, hence U + F ⋆ O F -invariant. (cid:3) Additional Integrals.
A consequence of Theorem 3.2 is that the function P Y : T ∗ ˆΣ → V ∗ E is a first integral of any function on L ∗ R pulled-back to T ∗ ˆΣby the Lax matrix L (equation (3.13)). Unfortunately, this map is not U + F -invariant.However, one is able to construct a map, f , from Y which is U + F -invariant. Naively,one might try to define f by means of equivariance. That is the task of this section.For each τ ∈ G F , let the subspace of V E spanned by { σ : σ | F = τ } be denotedby V τ,E . One may define Y τ = X σ | F = τ Y σ · ˆ σ, (3.36) NTROPY AND TODA LATTICES 17 for all τ ∈ G F . Since Y ¯ σ = ¯ Y σ for all σ , it is clear that complex conjugation inducesa real linear isomorphism between V ∗ τ,E and V ∗ ¯ τ,E . This linear isomorphism maps Y τ → Y ¯ τ , which implies that as real vector spacesV ∗ E ∼ = X τ ∈ B F V ∗ τ,E . (3.37)In the sequel, this natural isomorphism is understood.The group U + F acts on V ∗ E by u · ˆ σ = σ ( u ) − · ˆ σ . Since u ∈ F , this action is( c.f. equation 2.16) u · Y = X τ ∈ B F τ ( u ) − · Y τ . (3.38) Lemma 3.5.
Let V ∗ E, = { Y ∈ V ∗ E : ∀ τ ∈ B F , Y τ = 0 } . (3.39) The set V ∗ E, is U + F -invariant and V ∗ E, / U + F is a smooth manifold of dimensiondim V ∗ E .Proof. Inspection of equation (3.38) shows the invariance of V ∗ E, . To prove thatthe action of U + F is free and proper, define ˆ q : V ∗ E, → V o,F / LLL F byˆ q ( Y ) = X τ ∈ G F ln | Y τ | · τ − X τ ∈ G F ln | Y τ | · t mod LLL F , (3.40)where t = 1 | G F | X τ ∈ G F τ. From equation (3.38), one sees that u ∗ | Y τ | = − ln | τ ( u ) | + | Y τ | , so ˆ q is U + F -invariant,hence it defines a continuous map q : V ∗ E, / U + F → V o,F / LLL F . The action of U + F istherefore both free and proper, since q maps cosets onto cosets. (cid:3) Define a function g τ : T ∗ ˆΣ → R by g τ ( P ) = | Y τ | = X σ | F = τ | Y σ | . (3.41)These functions are first integrals of H (see below), but they are not U + F -invariant.However, their product is invariant:k = Y τ ∈ G F g τ = Y τ ∈ G F X σ | F = τ | Y σ | . (3.42)From k one obtains the important subspaces U = { P ∈ T ∗ Σ : k( P ) = 0 } , Z = { P ∈ T ∗ Σ : k( P ) = 0 } . (3.43)It is clear from the definition of k that Z is the union ∪ τ ∈ G F Z τ , where Z τ = g − τ (0)(although g τ is not U + F -invariant, its zero set is). It is also clear that U is an openand dense analytic submanifold of T ∗ Σ, Z = Σ × V ∗ o,F × V ∗ E, , and that Z is ananalytic sub-variety. Lemma 3.6.
Define the map f : U → V ∗ E, / U + F by, for all P ∈ U , f ( P ) = Y · U + F (3.44) where the action of U + F is given by equation (3.38). Then f is an analytic submer-sion.Proof. This is clear. (cid:3)
Remark 3.5. (1) U is a union of regular Liouville tori and singular tori (see below).The singular set Z has co-dimension equal to [ E : F ]. Therefore, when F ( E ,the co-dimension is two or more. In this case, the set of regular Liouville tori isconnected. (2) The topological structure of V ∗ E, / U + F is interesting. The map q isa submersion whose typical fibre is diffeomorphic to the Cartesian product of theunit spheres in V ∗ τ,E , for τ ∈ B F , with the positive real numbers. The bundle isgenerally non-trivial, since the action of U + F twists the fibres. Indeed, one sees thatthe map q × k is a proper submersion with Q τ ∈ B F S d τ − (cid:31) (cid:127) / / V ∗ E, / U + F q × k / / / / V o,F / LLL F × R + (3.45)where we identify k with a function defined on V ∗ E and d τ = [ E : F ]. This alsoexhibits V ∗ E, / U + F as a compact manifold times R + . The compact manifold issomething like a torus bundle over a torus. In particular, the ends of V ∗ E, / U + F are quite uncomplicated. (3) Let us relate the preceding discussion to that in theintroduction, c.f. diagram (1.5) and figure 1. Let ∼ be the equivalence relation onV ∗ E that is generated by defining Y ∼ Y τ = 0 for some τ ∈ B F and Y ∼ u · Y for all u ∈ U + F . The topological space V ∗ E / ∼ is a quotient of V ∗ E / U + F whereone collapses the set { Y : Q τ ∈ B F Y τ = 0 } to a point. We have the followingcommutative diagram:ˆ U Y / / i n c l. y y ssssssssssss / U + F (cid:15) (cid:15) V ∗ E, / U + F (cid:15) (cid:15) i n c l. y y sssssssssss T ∗ ˆΣ / U + F (cid:15) (cid:15) ˆ f = Y / / V ∗ E/ U + F (cid:15) (cid:15) U f / / i n c l. y y sssssssssssss V ∗ E, / U + F % % KKKKKKKKKK i n c l. % % KKKKKKKKKK i n c l. y y ssssssssss T ∗ Σ / / f V ∗ E / U + F collapse / / V ∗ E / ∼ (3.46)in (1.5) and ˆ f = Y is the momentum-map of the torus V E / N E acting on T ∗ ˆΣ.One can see that V ∗ E, / U + F is the complement of the coset of 0 in V ∗ E / ∼ andthat the first-integral map f is the natural extension of the map f from U to T ∗ Σ.From the diagram (3.45), one can see that V ∗ E / ∼ is homeomorphic to the coneon R + \ V ∗ E, / U + F , where R + acts by scalar multiplication. The diagram (3.45) alsoshows that when F = E , the fibres of q × k are disconnected, so that V ∗ E / ∼ isa union of disjoint cones pinched at the cone point as in figure (1). When F is aproper subfield of E , then the fibres of q × k are connected and V ∗ E / ∼ is a cone ona connected space. (4) There are globally-defined, U + F -invariant functions on V ∗ E .The most natural construction is a generalisation of the quadratic Casimir from the3-dimensional Sol manifolds and the Casimir in the totally real case [6, 11]. Eachcopy of V τ,E may be naturally identified with V
E/F , and similarly for the dualspaces. One can therefore define the map k ( Y ) = Y τ ∈ G F Y τ , k : V ∗ E → S d (V E/F ) (3.47) Note that V
E/E = R . NTROPY AND TODA LATTICES 19 where d = [ E : F ] and S ∗ (V E/F ) is the vector space of polynomial functions on thevector space V
E/F . It is clear that k is U + F invariant. A simple computation showsthat q × k is a submersion on V ∗ E, / U + F . (5) One might want to use the map Y X τ ∈ B F Y τ | Y τ | , V ∗ E, → Y τ ∈ B F S d τ − ⊂ X τ ∈ B F V ∗ τ,E to “split” the fibre bundle in (3.45). In general, however, the map induced by equiv-ariance is not well-defined. Rather, to obtain a well-defined map by equivariance,the set U τ = (cid:8) τ ( u ) / | τ ( u ) | : u ∈ U + F (cid:9) needs to be finite for all τ ∈ G F ; if one ofthese sets is not finite, then the co-domain of the induced map is not a manifold; ifall the sets are finite, then the induced map’s co-domain is a product of lens spacesso it does not split the fibre bundle, but it does split a suitable finite covering.Finiteness fails in many important cases: if F possesses a unit of infinite order onthe unit circle, for example. (6) The map f induces a sub-algebra of C ∞ ( T ∗ Σ) by R = (cid:8) f ∗ h : h ∈ C ∞ (V ∗ E, / U + F ) , h has compact support (cid:9) . (3.48)The sub-algebra R is the substitute on T ∗ Σ for the momentum map P Y on thelevel of algebras of functions.4. Complete Integrability
Let Z ∞ ( L ∗ ) be the set of smooth Casimirs of L ∗ with its standard Poissonbracket. This section proves that Theorem 4.1.
Let h ∈ Z ∞ ( L ∗ ) be a Casimir and let L : T ∗ Σ → L ∗ R be the Laxmatrix of equation (3.13) L ( P ) = X σ ∈ G E e r + Φ( X ) + X σ ∈ G E | Y σ | b σ · exp( b σ · h ˆ σ, x i ) · e − r , where Φ : V ∗ o,F → h ∗ satisfies the conclusions of Theorem (3.2). Then, the followingare true (1) H := L ∗ h is a completely integrable Hamiltonian with smooth integrals; (2) the algebras L := L ∗ Z ∞ ( L ∗ ) and R form a dual pair; (3) the singular set is an analytic variety.Proof. (1-2) Let ˆ R = ˆΠ ∗ R be the pullback of R to T ∗ ˆΣ. By the construction of R ,ˆ R ⊂ Y ∗ C ∞ (V ∗ E ) and their functional dimension is equal on ˆ U .A Casimir h of L ∗ is, a fortiori , invariant under the co-adjoint action of L .Therefore, h | L ∗− + L ∗ + L ∗ +1 must be functionally dependent on the co-adjoint invari-ants of L , e r · e − r , r ∈ ΨΨΨ and x ∈ L . From the formula for L , the function H = L ∗ h must therefore be a function of γ σ = | Y σ | b σ exp( b σ h ˆ σ, x i ) and X . Thesefunctions, and therefore H , are involution with ˆ R .This proves that L and R are commuting algebras of functions whose sum L + R is also abelian.Let R ⊂ e+ L ∗ + L ∗ +1 be the set of regular points of the algebra Z ∞ ( L ∗ ) restrictedto the subspace e + L ∗ + L ∗ +1 , where e = P r ∈ ΨΨΨ e r . This regular-point set is an openand dense real-analytic subset of e + L ∗ + L ∗ +1 . Since L | U is an analytic submersionwhose image is open in e + L ∗ + L ∗ +1 , L − ( R ) is an open and dense analytic subsetof U .Therefore, for all P ∈ L − ( R ),dim d L P = rank Ψ = dim V o,F , dim d R P = dim V E , while it is clear that d L P ∩ d R P = { } . Since dim Σ = dim V o,F + dim V E , this proves (1-2).(3) The singular set of R + L is the union of L − ( R c ) and Z = k − (0). Both arereal-analytic subsets of T ∗ Σ, hence their union is, too. (cid:3)
Theorem 1.
Let A be maximal in the sense of definition 1. This implies that Z b is an irreducible A -module. Let T ∈ GL( b ; C ) be a matrix that conjugates A toa subgroup of the set of diagonal matrices in GL( b ; C ) and let Γ = T − AT and M = T − Z b . Let F be the extension field of Q that is generated by the (1 , γ ∈ Γ; since A is maximal, F/ Q has degree b . The map δ , defined foreach γ ∈ Γ by, δ ( γ ) := γ δ : Γ → U F is a group homomorphism. Indeed, the maps δ j ( γ ) = γ jj are group homomorphismsinto the group of units of the j -th conjugate of F .It is clear that the first column of the matrix T can be supposed to have entriesin O F and the j -th column of T can be supposed to be the j -th conjugate of thefirst column. It is claimed that det T = q · d where q is a non-zero integer q and d isthe different of F . By definition, d = det U where the entries of the first column of U form a Z -basis of O F and the remaining columns are the conjugates of the firstcolumn. Let v be the first column of T . If the entries of v do not rationally span F ,then there is a non-zero t ∈ Z b such that h t, v i = 0. One can take the conjugatesof this linear equation and conclude that t is orthogonal to each column of T andtherefore t = 0. Absurd. One concludes that the entries of v generate a finite indexsubgroup of O F . The index of this subgroup is det T /d . This proves the claim.Therefore, for all m ∈ M , the j -th entry of qd × m lies in the j -th conjugate of O F .Define the map δ for each m ∈ M by δ ( m ) := qd · m δ : M → O F , where m is the first entry of m . It is clear that δ is a morphism of modules thatfaithfully intertwines the representation of Γ on M with that of δ (Γ) on δ ( M ); or, δ extends to a group embedding M ⋆ Γ (cid:31) (cid:127) / / O F ⋆ U F whence there is a groupembedding Z b ⋆ A (cid:31) (cid:127) / / O F ⋆ U F .Because Z b is an irreducible A -module, the degree of F/ Q is b so Z b is em-bedded as a finite index subgroup of O F . Since A is maximal, A is embedded asa torsion-free, finite-index subgroup of U F . Since δ (Γ) is torsion-free, there is achoice of U + F such that δ (Γ) ⊂ U + F . Therefore, one has obtained an embedding Z b ⋆ A (cid:31) (cid:127) / / O F ⋆ U + F which is of finite index. This proves that Σ A is a finitecovering of the manifold Σ constructed in lemma (2.3) with E = F .The proof of the theorem follows now by virtue of theorem 4.1 and the fact thatthe covering map T ∗ Σ A → T ∗ Σ is a local symplectomorphism. (cid:3)
Examples.
Let us illustrate the results of this section with two examples.4.1.1.
A non-normal cubic extension.
To illustrate the construction behind Theo-rem 1 take the case where A ⊳ GL(3; Z ) is the group generated by A = , A = − − − . (4.1) NTROPY AND TODA LATTICES 21 A is conjugate by a T ∈ SL(3; R ) to the group Γ generated by B = α α
00 0 α , B = α α
00 0 α , (4.2)where α j for j = 1 , , f ( x ) = x − x − α j = α j − − j = 4 , ,
6. For definiteness, one can take T to be the matrix T = α + α α + α α + α α + α α α , (4.3)whence det T = √ f and the number field F = Q [ α ].Let M = T − ( Z ) and ∆ = M ⋆
Γ so that T ∗ Σ A = T ∗ (∆ \ R × R ).To define the Lax matrix in (3.14), it is convenient to embed A by A log ◦ Ad T − / / h , A i +1 (cid:31) / / log | α i +1 | log | α i +2 | log | α i +3 | for i = 0 ,
1, where h ∼ = R is the Cartan subalgebra of SL(3; R ) consisting of tracezero diagonal 3 × A as a lattice in h . One can define thecoordinates for P = ( Y , y , X , x ) ∈ T ∗ ˜Σ = T ∗ R × T ∗ h and thereby obtain the Laxmatrix L ( P ) = λ − + X X
00 0 X + 12 × δ
00 0 δ λδ (4.4)where δ i = | Y i | exp(2 x i − x i +1 ) and P X i = P x i = 0. One obtains the twoPoisson-commuting functions H = 12 × Tr( L ) = 12 × (cid:0) X + X + X (cid:1) + 12 × ( δ + δ + δ ) (4.5) F = 13 × Tr( L ) ≡ × (cid:0) X + X + X (cid:1) − × ( δ X + δ X + δ X ) (4.6)that are in involution with Y .One may permute the indices i ; it is clear that a cyclic permutation yields thesame H and it is not difficult to see that transpositions yield equivalent hamiltonians(remark 7.2).4.1.2. A non-normal cubic extension and Z . To illustrate the construction behindTheorem 1 take the case where A ⊳ GL(6; Z ) is the group generated by A = − −
20 0 0 − − − − −
20 1 − − , A = − − − − − − − − − − −
20 1 − − − − . (4.7) A is conjugate by a T ∈ SL(6; R ) to the group Γ generated by B = α I α I
00 0 α I , B = α I α I
00 0 α I , (4.8)where I is the 2 × α j for j = 1 , , f ( x ) = x − x − α j = α j − − j = 4 , ,
6. One notes that the matrix A
12 LEO T. BUTLER is the matrix of the root α acting on the integers of O E , where E = Q [ α , √ F of the previous example. There is not a simpleexpression for such a matrix T , because unlike the previous example A is notconjugate over Z to its companion matrix. In all events, let M = T − ( Z ) and∆ = M ⋆
Γ so that T ∗ Σ A = T ∗ (∆ \ R × R ).To define the Lax matrix in (3.14), it is convenient to embed A by A log ◦ Ad T − / / h , A i +1 (cid:31) / / log | α i +1 | I log | α i +2 | I log | α i +3 | I consisting of trace zero diagonal 6 × A as a lattice in thesubspace h ⊂ h consisting of matrices which are of the form B ⊗ I for a 3 × B . One can define the coordinates for P = ( Y , y , X , x ) ∈ T ∗ ˜Σ = T ∗ R × T ∗ h and thereby obtain the Lax matrix L ( P ) = λ − I I I + X I X I
00 0 X I + 12 × δ I
00 0 δ I λδ I (4.9)where δ i = | Y i | exp(2 x i − x i +1 ) and P X i = P x i = 0. One obtains the twoPoisson-commuting functions H = 14 × Tr( L ) = 12 × (cid:0) X + X + X (cid:1) + 12 × ( δ + δ + δ ) (4.10) F = 16 × Tr( L ) ≡ × (cid:0) X + X + X (cid:1) − × ( δ X + δ X + δ X ) (4.11)that are in involution with Y ( ≡ indicates equality modulo functions of Y ).4.1.3. A non-normal quartic extension.
To illustrate the construction behind The-orem 1 take the case where A ⊳ GL(4; Z ) is the group generated by A = − − − , A = −
11 0 − . (4.12) A is conjugate by a T ∈ SL(4; R ) to the group Γ generated by B = diag( α , . . . , α ) , B = diag( α , . . . , α ) , (4.13)where α j for j = 1 , . . . , f ( x ) = x − x + x − x − α j = α j − − α j − − j = 5 , . . . ,
8. The roots α j equal12 × (cid:18) s √ t q (1 + s √ − (cid:19) where s, t ∈ {± } , j = 1 , . . . ,
4. This givestwo real reciprocal roots that are approximately 1 .
883 and 0 .
531 and two conjugatecomplex roots on the unit circle that are approximately 0 . ± . √−
1. Since f is Q -irreducible, the complex roots are not roots of unity, which also implies thatthe largest positive root is a Salem number. One notes that the matrix A is thematrix of the root α acting on the integers of O F , where F = Q [ α ].As with example 4.1.1, one can compute a straightforward representation of T − α − α + α − α α − α + α − α α − α + α − α α − α + α − α α − α + α − α α − α + α − α − α + α + α − α + α + α − α + α + α α + α α + α α + α (4.14) NTROPY AND TODA LATTICES 23 and one can verify that det T = − √−
7, which is the different of F . In all events,let M = T − ( Z ) and ∆ = M ⋆
Γ so that T ∗ Σ A = T ∗ (∆ \ R × R ).To define a Lax matrix as in (3.14), it is convenient to embed A into the Cartansubalgebra h ∼ = R of the real symplectic group of 4 × h = { diag( a, b, − a, − b ) : a, b ∈ R } by the embedding A log ◦ Ad T − / / h , A i +1 (cid:31) / / diag(log | α i +1 | , log | α i +2 | , log | α i +4 | , log | α i +3 | )for i = 0 ,
1. To make this an embedding, one must stipulate that the roots α j and α − j must be reciprocals for j = 1 ,
2; it is also supposed that α (resp. α ) haspositive imaginary part (resp. is the largest real root of f ). This embeds A as alattice in h . One can define the coordinates for P = ( Y , y , X , x ) ∈ T ∗ ˜Σ = T ∗ R × T ∗ h and thereby obtain the Lax matrix L ( P ) λ − − + Φ( X ) z }| { a X a X − a X
00 0 0 − a X + 12 × λδ δ − δ δ (4.15)where δ i , a i are determined in Table 2. One obtains the two Poisson-commutingfunctions H = 14 × Tr( L ) = 12 × (cid:0) a X + a X (cid:1) + 12 × ( δ + 12 δ + 12 δ ) (4.16) F = det L ≡ δ δ δ − δ a a X X + a X δ a X δ a a X X (4.17)that are in involution with Y ( ≡ indicates equality modulo functions of Y ).To explain the following choices for the functions δ i , one defines the embeddingsof the number field F by τ i ( α ) = α i , so that B F = { τ , τ , τ } and τ = ¯ τ .A bijection ρ : B F → ΨΨΨ is identified as a permutation s of { , , } under theconvention that ρ ( τ i ) = r s ( j ) . Only three choices are listed since the remainingthree are obtained by permuting Y and Y in the formulae below (these unlistedchoices are also conjugate to the listed choices, since this permutation induces ananalytic symplectomorphism of T ∗ Σ). c ρ b τ a , a δ δ δ , , , | Y | e x − x | Y | e x | Y | e − x , , , | Y | e x − x | Y | e x | Y | e − x , , , | Y | e x − x | Y | e x | Y | e − x Table 2.
Choices for the Lax matrix L ; y i ( Y i ) is a coordinate onthe α i -eigenspace with y = ¯ y ( Y = ¯ Y ). See Theorem 3.2.With these choices of δ i , (4.15) gives a Lax representation of the hamiltonianvector field of H (4.16) with the integral F (4.17). Although fibrewise convex forall choices, the hamiltonian H is only fibre-wise quadratic for the first choice. Additional Lax Representations.
One can define additional Lax representations withthe aid of the remaining rank 2 affine Kac-Moody algebras. A (1)2 . Embed A into the Cartan subalgebra h ∼ = R of SL(3; R ) via A log ◦ Ad T − / / h , A i +1 (cid:31) / / diag(2 log | α i +1 | , log | α i +2 | , log | α i +3 | )where the roots α j are labelled as above. This embeds A as a lattice in h . One candefine the coordinates for P = ( Y , y , X , x ) ∈ T ∗ ˜Σ = T ∗ R × T ∗ h and thereby obtainthe Lax matrix L ( P ) = λ − + a X a X
00 0 a X + 12 × δ
00 0 δ λδ (4.18)where P a i X i = P a − i x i = 0 and δ i is defined below. One obtains the two Poisson-commuting functions H = × Tr( L ) and F = det L where H = a X + a X a X + a X + 12 × ( δ + δ + δ ) (4.19) F ≡ − a X a X − a X a X + 12 a X ( δ − δ ) + 12 a X ( δ − δ ) , (4.20)where the functions δ i are determined in table 3, following the conventions in table2. c ρ b τ a , a δ δ δ , , , | Y | e x − x | Y | e − x − x | Y | e x +8 x , , , | Y | e x − x | Y | e − x − x | Y | e x +4 x , , , | Y | e x − x | Y | e − x − x | Y | e x +4 x Table 3.
Choices for the Lax matrix L ; y i ( Y i ) is a coordinate onthe α i -eigenspace with y = ¯ y ( Y = ¯ Y ). See Theorem 3.2. G (1)2 . One proceeds as above and obtains the hamiltonian H = 124 × (cid:0) a X + 3 a X a X + 3 a X (cid:1) + 16 × (3 δ + δ + δ ) (4.21)where δ i is defined by c ρ b τ a , a δ δ δ
12 (1) 8 , , , | Y | e x − x | Y | e x − x | Y | e − x
12 (3 2) 8 , , , | Y | e x − x | Y | e x − x | Y | e − x
24 (3 2 1) 24 , , , | Y | e x − x | Y | e x − x | Y | e − x
12 (2 1) 6 , , , | Y | e x − x | Y | e x − x | Y | e − x
12 (2 3 1) 6 , , , | Y | e x − x | Y | e x − x | Y | e − x
24 (3 1) 24 , , , | Y | e x − x | Y | e x − x | Y | e − x Table 4.
Choices for the Lax matrix L ; y i ( Y i ) is a coordinate onthe α i -eigenspace with y = ¯ y ( Y = ¯ Y ) and x i is the coordinateon h induced by the simple coroots [17, p. 346]. NTROPY AND TODA LATTICES 25 The singular set and gradient flows
Two prefatory comments: first, the fibre bundle structureV E / O E (cid:31) (cid:127) / / Σ p / / / / V o,F / LLL F induces the sub-bundle V = ker dp ⊂ T Σ and its annihilator V ⊥ ⊂ T ∗ Σ. Thesub-bundle V ⊥ is naturally isomorphic to Σ × V ∗ o,F . Second, recall that the stablemanifold of a point p is the set of points whose orbits converge to that of p ’s astime goes to ∞ ; the unstable manifold is defined symmetrically as time goes to −∞ ;the stable and unstable manifolds of a set are the union of the stable and unstablemanifolds of each point in the set. In this section it is shown that Theorem 5.1. V ⊥ is an invariant set for the Hamiltonian flow of H (equation3.16). The stable and unstable manifolds of V ⊥ , W ± ( V ⊥ ) , coincide and W ± ( V ⊥ ) = Z (= k − (0)) . (5.1)Before proceeding with the proof, let us explain why theorem 5.1 is natural fromthe perspective of Bogoyavlenskij-Toda lattices. It is a well-known result that theopen Bogoyavlenskij-Toda lattices undergo scattering: the particles interact oversome time interval and then separate and proceed off to infinity. The net result ofthe interaction is that the momenta of the particles may be permuted from t = −∞ to t = ∞ ; in terms of the Lax matrix, L ( −∞ ) and L ( ∞ ) are diagonal matriceswhich differ by the action of some element in the Weyl group. Since the openBogoyavlenskij-Toda lattices are obtained from the periodic Bogoyavlenskij-Todalattices by turning off the potential term associated to the root ηηη , it is plausiblethat when other potential terms are turned off, the system should still exhibit suchscattering behaviour. To confirm this, one must develop the double-bracket orgradient representation of these systems.5.1. Double-bracket and gradient representations.
Let us recall the construc-tions of [9], where it is demonstrated that the open Bogoyavlenskij-Toda latticesmay be viewed as gradient flows. Let g be a semi-simple Lie algebra with Cartan-Killing form κ = hh , ii . For x ∈ g ∗ let O x denote the co-adjoint orbit of x , let g x bethe stabiliser algebra of x and let g ⊥ x be the κ -orthogonal complement of g x . Themap v ad v x is a linear isomorphism of g ⊥ x with T x O x . Definition 5.1.
The normal metric , n , on O x is defined at T x O x by ∀ u, v ∈ g ⊥ x : n (ad u x, ad v x ) = hh u, v ii (5.2) Lemma 5.2. If H ∈ C ∞ ( g ∗ ) , then the gradient vector field of H |O x at x is n ∇ H ( x ) = − [ x, [ x, y ]] ∈ T x O x (5.3) where y = ∇ H ( x ) is the κ -gradient of H . For a proof, see [9].5.2.
Bogoyavlenskij-Toda lattices and double brackets.
Let us specialise theconstruction of the previous section. The semi-simple Lie algebra is the loop algebra L or its twisted counterpart of section 3.1. Let ΨΨΨ ( ΨΨΨ be a proper subset obtainedby removing a single root from ΨΨΨ. Let x = h + X r ∈ ΨΨΨ x r (e r + e − r ) ∈ L ∗ , m = X r ∈ ΨΨΨ x r (e r − e − r ) , X ( x ) = [ x, m ] , (5.4)where h ∈ h . The vector field X is a Bogoyavlenskij-Toda-like vector field associatedto the splitting of L ⊂ L as in section 3.2.
Lemma 5.3. X is a gradient vector field relative to the normal metric, hence X is tangent to O x .Proof. It suffices to determine a y ∈ h such that X = n ∇ H where H ( x ) = hh x, y ii .To do so, it suffices to determine y such that m = − [ x, y ]. This reduces to thesolubility of the equations ∀ r ∈ ΨΨΨ : x r = x r h r, y i . (5.5)Since at least one of the x r vanishes, and any subset of ΨΨΨ of cardinality − h ∗ , there is always a solution to (5.5). (cid:3) The vector field X is equivalent to the differential equations − ˙ h = X r ∈ ΨΨΨ x r h r , and ∀ r ∈ ΨΨΨ : ˙ x r = x r h r, h i , (5.6)where h r = [e r , e − r ]. In particular, X is tangent to x r = 0 for any r . It is also clearthat X vanishes at x iff ∀ s ∈ ΨΨΨ : x s h s, h i = 0 and X r ∈ ΨΨΨ x r hh s, r ii = 0 , (5.7)where the identity h s, h r i = hh s, r ii has been used. Since the matrix [ hh s, r ii ] r,s ∈ ΨΨΨ has full rank, the second part of (5.7) implies that x r = 0 for all r ∈ ΨΨΨ andtherefore for all r ∈ ΨΨΨ. This proves that
Lemma 5.4. X vanishes at x iff x ∈ h . It remains to prove that all orbits of X limit onto h . Since ˙ H = hh y, − [ x, [ x, y ]] ii = hh ad y x, ad y x ii , and ad y x = − P r ∈ ΨΨΨ x r h r, y i (e r − e − r ) one concludes from (5.5)that˙ H = − X r ∈ ΨΨΨ x r ≤ X = 0. Thus, the ω -limit set of every point x lies in h , hence O x ∩ h .The latter is a finite set and since X is a gradient vector field on O x , the ω -limitset is a single point. Let h ∈ O x ∩ h be this point and let ΨΨΨ = { r : x r = 0 } . Letus linearise X about h subject to the condition that x r = 0 for all r ∈ ΨΨΨ : − δ ˙ h = 0 , and ∀ r ΨΨΨ : δ ˙ x r = δx r h r, h i , (5.9)where δx, δh denote variations. It is clear that a necessary condition for stability of h is that h r, h i ≤ r ΨΨΨ . A simple argument involving the transitivityof the action of the Weyl group on the Weyl chambers, shows that such an h mustexist. This proves Lemma 5.5.
For each x of the form in equation (5.4), the ω -limit set of x underthe gradient flow of X = n ∇ H is a point h ∈ O x ∩ h that satisfies h r, h i ≤ forall r ∈ ΨΨΨ . A similar statement is true for the α -limit set, too. It should be observed thatwhile h contains the ω -limit set of every point x , h is not a normally hyperbolicmanifold. One can see this from (5.9): when h h , r i = 0, one loses hyperbolicity. Theorem 5.1.
For each τ ∈ G F the Hamiltonian vector field of H in (3.16), whenrestricted to the invariant set g − τ (0) (equation 3.41), is semi-conjugate to a vectorfield of the form of X in (5.4). The semi-conjugacy is provided by the Lax repre-sentation in equation (3.15). Lemma 5.5 implies that the ω -limit set of a point P ∈ g − τ (0) lies in V ⊥ . Similarly for the α -limit set of P . Since k − (0) = ∪ τ ∈ G F g − τ (0),this proves the theorem. (cid:3) NTROPY AND TODA LATTICES 27 Uniqueness up to Energy-Preserving Topological Conjugacy
Marked Homology Spectrum of a Flow.
Two flows φ : M × R → M and ϕ : N × R → N are topologically conjugate if there is a homeomorphism h : M → N such that hφ t = ϕ t h for all t ∈ R . Let P φ be the set of periodicpoints of the flow φ . For each periodic orbit γ of φ , let the homology class of γ bedenoted by ¯ γ and its period by Period( γ ). Let P φ, ¯ γ,T denote the union of periodicorbits of φ whose homology class is ¯ γ and period is T . The number of connectedcomponents of P φ, ¯ γ,T is denoted by β φ, ¯ γ,T . The following two definitions originatein Schwartzman’s work [28]. Definition 4.
Let M φ = { (¯ γ, Period( γ ) , β φ, ¯ γ, Period( γ ) ) : γ ∈ P φ } . We call M φ the marked homology spectrum of φ . The marked homology spectrum is a subset of H ( M ; Z ) × R × N that is an in-variant of topological conjugacy in the following sense: if φ and ϕ are topologicallyconjugate then ( h ∗ × id R × id N )( M φ ) = M ϕ , where h ∗ : H ( M ; Z ) → H ( N ; Z ) is the obvious isomorphism. Example 6.1 . Let v ∈ V o,F and define the flow φ v : Σ × R → Σ by φ vt ( y , x ) = ( y , x + tv ) mod ∆ . (6.1)A point ( y , x ) ∈ Σ is periodic of period T for φ v iff T v = ℓ ( u ) for some u ∈ U + F and u · y = y mod N E .The map u : V E / N E → V E / N E is a toral automorphism. The number of fixedpoints of u is, up to sign, the degree of the map u −
1. The latter is det( u −
1) = Q σ ∈ G E σ ( u − u − ∈ E . But since u − ∈ F , thisnorm equals N F ( u − [ E : F ] .Thus, M φ v = n ( ℓ ( u ) , T, | N F ( u − | [ E : F ] ) : ∀ u ∈ U + F & T ∈ R + s . t . T v = ℓ ( u ) o (6.2) Example 6.2 . Let Q : V ∗ o,F → V o,F be a linear isomorphism and M = T ∗ (V o,F / LLL F ) =V ∗ o,F × V o,F / LLL F . Let φ t ( X , x ) = ( X , x + t Q· X mod LLL F ). Clearly, η ± ( X , x ) = {±Q· X } for all ( X , x ) ∈ M .Let V = { ( X , x ) ∈ M : hQ · X , X i = 1 } be the unit-sphere bundle, φ = φ |V and | m | Q = p |hQ − m, m i| for all m ∈ V o,F . The marked homology spectrum of φ is easily seen to equal M φ = { ( ℓ ( u ) , | ℓ ( u ) | Q ,
1) : u ∈ U + F } . (6.3) Example 6.3 . The fibre-bundle structure V E / N E (cid:31) (cid:127) / / Σ p / / / / V o,F / LLL F allowsone to pullback the unit-sphere bundle V and the flow φ of the previous example.Let ϕ be the pulled-back flow on p ∗ V . The previous two examples show that themarked homology spectrum of ϕ is M φ = { ( ℓ ( u ) , | ℓ ( u ) | Q , | N F ( u − | [ E : F ] ) : u ∈ U + F } . (6.4)The marked homology spectrum is especially interesting because it contains in-formation about both the quadratic form restricted to the Dirichlet lattice, andit contains information about the periodic points of the toral automorphisms u :V E / N E → V E / N E for u ∈ U + F . In [11], this extra information about the fixed points of the toral automorphisms was not noticed. It turns out that this informa-tion is extremely important.6.2. Asymptotic Homology of a Flow.
Let π : ˆ M → M be the universal abeliancovering of M . The flow φ is covered by a flow ˆ φ : ˆ M × R → ˆ M . Let F ⊂ ˆ M bea fundamental domain for the group of deck transformations Deck( π ). For each p ∈ M , choose ˆ p ∈ F ∩ π − ( p ). For each t there is a g ∈ Deck( π ) such thatˆ φ t ( p ) ∈ g.F ; let g t ( p ) be one such element and let t g t ( p ) ∈ Deck( π ) ⊗ Z R . Recallthat Deck( π ) ⊗ Z R ≃ H ( M ; R ). Definition 5.
Let η φ ( p ) := \ T ≥ (cid:26) t g t ( p ) : t ≥ T (cid:27) be the asymptotic homology of p ∈ M . Let η ± φ = η φ ± where φ ± t = φ ± t . One can show that η φ ( p ) is independent of the choice of representatives and if M is compact then η φ ( p ) is non-empty for all p . It is also clear that if there is asemi-conjugacy h with h ◦ φ = ϕ ◦ h , then h ∗ η ± φ ( p ) = η ± ϕ ( h ( p )). Lemma 6.1.
Let H be a Hamiltonian defined by equation 3.16, and let ϕ : T ∗ Σ × R → T ∗ Σ be its Hamiltonian flow. Let U τ = { g τ = 0 } for each τ ∈ G F . If P ∈ U τ ,then h η ± ϕ ( P ) , ˆ τ i ≤ . Remark.
This lemma is very close in spririt to lemma 5.5.
Proof.
Let ˆ P = ( Y , y + N E , X , x ) ∈ ˆ U σ and let P = Π( ˆ P ), c.f. (2.15). Since g τ ( ˆ P ) =0, Y τ = 0. If v ∈ η ± ϕ ( P ), then there is a sequence T k → ±∞ such that v = lim k →∞ | T k | ( x ( T k ) − x (0)) , where ˆ ϕ t ( y + N E , Y , X , x ) = ( Y ( t ) , y ( t ) + N E , X ( t ) , x ( t )) and ˆ ϕ t is the lift of ϕ t to T ∗ ˆΣ. Thus: h v, ˆ τ i = lim k →∞ | T k | h x ( T k ) , ˆ τ i . On the other hand ˆ H and g τ are first integrals of ˆ ϕ t . Inspection of equation 3.16shows that ˆ H ( ˆ P ) ≥ g b τ / τ exp( b σ h x ( T ) , ˆ τ i ) for all T . Since b σ , b τ > g τ = 0,this inequality implies that1 | T k | h x ( T k ) , ˆ τ i ≤ | T k | b σ (cid:18) ln ˆ H − b τ g τ (cid:19) k →∞ −→ . Since v ∈ η ± ϕ ( P ) was arbitrary, this proves the lemma. (cid:3) As noted above, the fibre bundle structure V E / N E (cid:31) (cid:127) / / Σ p / / / / V o,F / LLL F ofΣ induces the sub-bundle V = ker dp ⊂ T Σ and its annihilator V ⊥ ⊂ T ∗ Σ. Thesub-bundle V ⊥ is the intersection of Z τ = g − τ (0) over all τ ∈ B F ; it is alsoisomorphic to Σ × V ∗ o,F . Lemma 6.2.
Let H , H be defined by equation 3.16 with root bases Ψ , Ψ . If h : T ∗ Σ → T ∗ Σ conjugates their Hamiltonian flows, then h ( V ⊥ ) = V ⊥ . NTROPY AND TODA LATTICES 29
Proof.
Let U be the set of points in V ⊥ that are mapped out of V ⊥ under h . Since P V ⊥ iff ∃ τ ∈ B F such that g τ ( P ) = 0, one sees that U = h − ( ∪ τ ∈ B F U τ ) ∩ V ⊥ . It suffices to prove that U is empty, since a symmetric argument applies to h − .Therefore, it suffices to prove that U τ = h − ( U τ ) ∩ V ⊥ is empty for all τ . Since U τ is open, U τ is an open subset of V ⊥ , so to prove that it is empty, it suffices to showthat U τ is nowhere dense. As noted above, V ⊥ is naturally isomorphic to Σ × V ∗ o .Let π o : V ⊥ → V ∗ o denote the projection onto the second factor. Clearly, π o is anopen map and π o ( P ) = X where P = Π(0 , y , X , x ) ∈ V ⊥ . It suffices to show that π o ( U τ ) lies in a hyper-plane to prove the lemma.Let ϕ i be the Hamiltonian flow of H i , and Q i the quadratic form used to define H i (Equation 3.16). If P ∈ U τ , then P ∈ V ⊥ so η ± ϕ ( P ) = {±Q · X } , while h ( P ) ∈ U τ , so from the previous lemma h η ± φ ( h ( P )) , ˆ τ i ≤ . Since ϕ t h = hϕ t , η ± ϕ ( h ( P )) = h ∗ η ± ϕ ( P )which implies that ±h h ∗ Q X , ˆ τ i ≤ . Therefore, h h ∗ Q X , ˆ τ i vanishes. Since h ∗ Q is non-degenerate, X = π o ( P ) lies in afixed hyper-plane. Thus, π o ( U τ ) lies in a hyper-plane. Since π o is an open map, U τ is empty. (cid:3) Remark 6.1.
Lemmas 6.1 and 6.2 can be reformulated and shown to hold inmuch greater generality. Let Σ A be defined as in 1.1 and let H : T ∗ Σ A → R be a smooth, fibre-wise convex hamiltonian that is left-invariant. Left-invarianceimplies that H enjoys the integral f (1.5). In particular, if one defines the function γ i ( P ) = | p y i exp( h ℓ i , x i ) | , then the properness of H implies that there is a function c = c ( H ) such that 0 ≤ γ i ( P ) ≤ c ( H ( P )) for all P ∈ T ∗ Σ A . The proof of lemma6.2 applies to show that if γ i ( P ) = 0, then h ℓ i , v i ≤ v ∈ η ± ( P ). Thisimplies that the asymptotic homology of a point P with Q i γ i ( P ) = 0 is trivial andthat a topological conjugacy of two such hamiltonian flows must map V ⊥ to itself. Definition 6.
A homeomorphism h : T ∗ Σ → T ∗ Σ is energy-preserving if h ( { H = } ) = { H = } . We use the notation of Lemma 6.2 and its proof:
Theorem 6.3.
Let H , H be defined by Equation 3.16 corresponding to root bases Ψ , Ψ . If h ∈ Homeo( T ∗ Σ) is an energy-preserving conjugacy of ϕ with ϕ , then (1) h ∗ : H ( T ∗ Σ) → H ( T ∗ Σ) induces automorphisms of LLL F and U + F such thatthe following commutes U + F α / / ℓ (cid:15) (cid:15) U + Fℓ (cid:15) (cid:15) LLL
F f / / LLL F ; ( ∗ )(2) f is an isometry of ( LLL F , Q ) with ( LLL F , Q ) ; (3) α preserves the number of fixed points of u ∈ U + F acting on V E / N E : | N F ( α ( u ) − | = | N F ( u − | ∀ u ∈ U + F . Proof. (1) The map h ∗ on H induces an automorphism f of LLL F . The isomorphism ℓ allows the definition of α as an automorphism of U + F and shows that (*) commutes.(2) Let V i = V ⊥ ∩ H − i ( ). Since h is energy preserving, Lemma 6.1 implies that h ( V ) = V . Let ϕ i |V i be denoted by Φ i and let h |V continue to be denoted by h .Examples 6.1 and 6.1 show that M Φ i = { ( ℓ ( u ) , | ℓ ( u ) | Q i , | N F ( u − | [ E : F ] ) : u ∈ U + F , u = ± } for i = 1 , h Φ equals Φ h , so from the identity M Φ = ( h ∗ × id R × id N ) M Φ one sees that | ℓ ( u ) | Q = | f ◦ ℓ ( u ) | Q = | ℓ ( α ( u )) | Q , (6.5) | N F ( u − | = | N F ( α ( u ) − | (6.6)for all u ∈ U + F . Equation (6.5) shows that f is an isometry, while equation (6.6)shows that α preserves the number of fixed points. (cid:3) Let us dualise Theorem 6.3. Let φ i be a linear isomorphism V ∗ o,F → h ∗ i inducedby a bijection ρ i : B F → Ψ (see Definition 3). The norms |·| Q i on LLL F are equivalentmodulo Aut( LLL F ) iff the dual norms | · | ∗Q i on LLL ∗ F are equivalent modulo Aut( LLL ∗ F ).Since, by Theorem 3.2, there is a c i ∈ N such that | X | ∗Q i = c − i p hh φ i ( X ) , φ i ( X ) ii i ,Theorem 6.3 implies Corollary 6.4. If ϕ and ϕ are topologically conjugate by an energy-preservinghomeomorphism, then there exists µ ∈ Isom( h ∗ ; h ∗ ) and g = f ∗ ∈ Aut(
LLL ∗ F ) suchthat µ = c c × φ gφ − . (6.7) Remark 6.2.
One might attempt to use Corollary 6.4 to try to determine thetopological conjugacy classes of Hamiltonian flows. This is the approach takenin [11]. However, this approach leads to some very delicate and long-outstandingissues in transcendence and algebraic-independence theory. This paper skirts thosedifficulties by employing all the information in the marked homology spectrum.6.3.
Periodic points of toral automorphisms.
Part (3) of Theorem 6.3 has auseful corollary: the number of period- k periodic points of the automorphisms u and α ( u ) of the torus V E / N E are equal for all k . Therefore, their asymptotic ratesof growth are equal. Define the function h : V o,F → R byh( v ) = X τ ∈ B F n τ h ˆ τ , v i + (6.8)for all v ∈ V o,F , where • + = max( • , k periodic points of u ∈ U + F is [ E : F ] × h( ℓ ( u )), this proves Lemma 6.5.
Under the hypotheses of Theorem 6.3, the automorphism f : LLL F → LLL F satisfies h = h ◦ f. The function h is piecewise linear. One can characterise the sets on which h islinear as follows. For J ⊂ B F , letV Jo,F := { v ∈ V o,F : ∀ τ ∈ J, h ˆ τ , v i > ∀ τ J, h ˆ τ , v i < } . (6.9)Note that if J = ∅ or J = B F , then V Jo,F is empty; otherwise V
Jo,F is an open setthat is closed under addition and multiplication by positive scalars. Since V
Jo,F is NTROPY AND TODA LATTICES 31 open and it is closed under positive dilations, it contains balls of arbitrarily largediameter and hence it contains points in
LLL F . Therefore, LLL JF := LLL F ∩ V Jo,F (6.10)is a non-empty subset of
LLL F , for all J ⊂ B F with J = ∅ and B F .To return to h: for all J ⊂ B F , define r J := X τ ∈ J n τ ˆ τ . (6.11) Lemma 6.6.
The following is true: (1) if v ∈ V Jo,F , then h( v ) = h r J , v i ; (2) if v ∈ LLL JF and f ( v ) ∈ LLL IF , then r J = f ∗ r I ; (3) for each J ⊂ B F with J = ∅ and B F , there is a unique I ⊂ B F such that f ( LLL JF ) ⊂ LLL IF ; (4) f induces a permutation πππ of the power set B F that satisfies (a) πππ ( ∅ ) = ∅ and πππ ( B F ) = B F ; (b) πππ ( J ) = I iff f ( LLL JF ) ⊂ LLL IF .Proof. (1) h may be characterised as: h( v ) = max I ⊂ B F h r I , v i . On the set V Jo,F ,this maximum is achieved uniquely at I = J . This proves that h = r J on V Jo,F .(2) Let v ∈ LLL JF and f ( v ) ∈ LLL IF . Lemma (6.5) implies that h r J , v i = h( v ) = h( f ( v )) = h f ∗ r I , v i . It is clear that the set
LLL JF ∩ f − ( LLL IF ) is an intersection of Zariski dense subsets ofV o,F , hence is Zariski dense since it is non-empty. Therefore r J must equal f ∗ r I on V o,F .(3) Let v i ∈ LLL JF and assume that f ( v i ) ∈ LLL I i F . Therefore, from the previous step f ∗ r I = r J = f ∗ r I . Since f is an automorphism r I = r I . Since the map I r I : B F → V ∗ o,F is injective except at ∅ and B F (both are sent to 0), oneconcludes that I = I .(4) From step (3), the properties (a-b) uniquely define a map πππ : B F → B F because V Jo,F = ∅ —hence LLL JF = ∅ — for all J ⊂ B F , J = ∅ , B F . This map πππ isinvertible because f is induced by the homeomorphism h : one can equally startwith h − , get f − and define πππ ′ thusly. Step (3) shows that πππ ′ = πππ − . (cid:3) Let us be more precise about the nature of f . Lemma 6.7 should be comparedwith [11, Theorem 7], where the Gel’fond conjecture [19] is invoked to obtain theweaker conclusion that f ∗ ∈ Aut(
LLL ∗ F ) ∩ Aut(V ∗ o,F, Q ) . Lemma 6.7.
Let V ∗ o,F, Z be the Z -module spanned by (cid:8) n τ ˆ τ | V o,F : τ ∈ B F (cid:9) and LLL ∗ F = Hom( LLL F , Z ) . Then f ∗ ∈ Aut(
LLL ∗ F ) ∩ Aut(V ∗ o,F, Z ) (6.12) Proof.
Note that πππ is defined by r J = f ∗ r πππ ( J ) for all J . If J = πππ − { τ } , then f ∗ ( n τ ˆ τ ) = r J ∈ V ∗ o,F, Z , (6.13)since r { τ } = n τ ˆ τ . On the other hand, if J = { τ } , then( f ∗ ) − ( n τ ˆ τ ) = r πππ ( J ) ∈ V ∗ o,F, Z . This proves that f ∗ ∈ Aut(V ∗ o,F, Z ), and since f ∈ Aut(
LLL F ), the lemma is proven. (cid:3) Lemma 6.8.
Let πππ : B F → B F be the permutation defined in Lemma 6.6. If I, J ⊂ B F are disjoint sets, then πππ ( I ⊔ J ) = πππ ( I ) ⊔ πππ ( J ) , ⊔ = disjoint union. Consequently, πππ is induced by a permutation of B F .Proof. Since I ∩ J = ∅ , r I ⊔ J = r I + r J . Therefore r πππ ( I ⊔ J ) = ( f ∗ ) − r I ⊔ J = ( f ∗ ) − ( r I + r J ) = r πππ ( I ) + r πππ ( J ) . (6.14)Assume that πππ ( I ) and πππ ( J ) are not disjoint. Then, there is a τ ∈ πππ ( I ) ∩ πππ ( J ). Thecoefficient on ˆ τ in the right-hand side of (6.14) is therefore 2 n τ . The coefficienton ˆ τ in the left-hand side of (6.14) is at most n τ , however. Absurd. Therefore πππ ( I ) ∩ πππ ( J ) must be empty.Consider the B F + 1 subsets of B F that contain at most 1 element. This isthe largest family of pairwise disjoint subsets of B F . Therefore, πππ must be mapthis family to itself. Since πππ ( ∅ ) = ∅ , πππ maps the singleton sets to singletons. (cid:3) Let the permutation of B F induced by πππ be denoted by πππ , too. Equation (6.13)is thereby simplified to ∀ τ ∈ B F : n σ f ∗ ˆ σ = n τ ˆ τ ⇐⇒ πππ ( τ ) = σ. (6.15)Intuitively, one wants to say that πππ should not mix up the real and non-real em-beddings, so the coefficients on both sides of (6.15) ought to be equal. To provethis, observe that (6.15) implies that ∀ u ∈ U + F : f ◦ ℓ ( u ) = X τ ∈ G F n τ n πππ ( τ ) ln | τ ( u ) | · τ. (6.16)Since f ∈ Aut(
LLL F ), the right-hand side lies in LLL F ⊂ V o,F for all u . Let ξ = P τ ∈ G F n τ n πππ ( τ ) ˆ τ ∈ V ∗ F ; one sees that h ξ, ℓ ( u ) i = 0 since f ◦ ℓ ( u ) ∈ V o,F . Since LLL F spans V o,F , this shows that ξ ∈ V ⊥ o,F . SinceV ⊥ o,F = span ( τ − ˆ τ , ǫ : τ ∈ G cF , ǫ = X τ ∈ G F ˆ τ ) , (6.17)and the coefficients n τ /n πππ ( τ ) are constant under the involution τ ¯ τ , one sees that ξ must be a multiple of ǫ . Therefore, n τ /n πππ ( τ ) must be independent of τ . Since πππ is a permutation, this forces n τ /n πππ ( τ ) to be identically equal to unity. This proves Lemma 6.9.
The permutation πππ of G F preserves the type of each embedding. Inparticular, ∀ τ ∈ B F : f ∗ ˆ σ = ˆ τ ⇐⇒ πππ ( τ ) = σ. (6.18) Lemma 6.10.
For each τ ∈ B F , there exists a homomorphism ζ τ : U + F → S suchthat (1) for all u ∈ U + F , τ ( α ( u )) = ζ τ ( u ) · σ ( u ) where πππ ( σ ) = τ ; (2) ζ τ maps U + F into S ∩ U K where K is the normal closure of F .Proof. The equation f ( ℓ ( u )) = ℓ ( α ( u )) implies, via equation (6.18), that | τ ( α ( u )) | = | σ ( u ) | when σ = πππ − ( τ ). Therefore, there is a unit modulus number ζ = ζ τ ( u ) suchthat τ ( α ( u )) = ζ · σ ( u ). The number ζ is a ratio of numbers in conjugates of F ,hence it lies in the smallest field containing all conjugates of F , K . Moreover, onesees that ζ τ is a ratio of two homomorphisms, hence it is a homomorphism. Finally,since ζ is a ratio of units of K , it is a unit of K . (cid:3) NTROPY AND TODA LATTICES 33
Strictly Hyperbolic Number Fields.
Lemma 6.10 shows that, if one canforce ζ τ to be trivial, then α is an automorphism of F and πππ is induced by rightcomposition by α − . One expects that this is always the case: the symmetries ofthe number field F/ Q ought to appear as symmetries (=topological conjugacies) ofthe Hamiltonian system, and vice versa. However, when K contains infinite orderelements in S , it is difficult to say anything meaningful about ζ τ . This is quitelikely related to the fact that if u ∈ U K ∩ S has infinite order, then the inducedautomorphism of the torus V K / O K is partially hyperbolic . Definition 7.
A unit u ∈ U F is hyperbolic if none of its conjugates have unitmodulus. F is hyperbolic if its only non-hyperbolic units are roots of unity. F is strictly hyperbolic if its normal closure, K , is hyperbolic. In other words, F is hyperbolic iff U F ∩ S < ∞ . (6.19)If F is hyperbolic, then U + F acts on the torus V F / N F as a group of Anosov au-tomorphisms; if F is strictly hyperbolic , then the ‘closure’ of U + F , U + K , acts on thetorus V K / O K as a group of Anosov automorphisms.Strict hyperbolicity is a property of the normal closure K : K itself is strictlyhyperbolic and so, therefore, are all its subfields. Examples of strictly hyperbolicnumber fields are legion; there also appear to be many hyperbolic but not strictlyhyperbolic number fields. Examples. (1) F is totally real if all its conjugates are real. In this case, its normal closure isalso totally real and so U K ∩ S = {± } . Thus, all totally real number fields arestrictly hyperbolic.(2) Let ζ be a p -th root of unity for some odd prime p . The field K = Q ( ζ ) hasthe totally real subfield F = Q ( ζ + ζ − ) of index 2. The Dirichlet theorem on thegroup of units implies that U F is of finite index in U K . Since F is totally real, K isstrictly hyperbolic.(3) More generally, let K/ Q be a non-real, normal extension of Q . If K has atotally real subfield F of index 2, then, as above, U F is a finite-index subgroup of U K , hence K is strictly hyperbolic.(4) A penultimate, concrete example: let F = Q ( a ) where a is the unique realroot of p ( x ) = x + 3 x −
1. The discriminant of p is d = − ×
5, so √ d Q ,which implies that F is not a normal extension of Q ( p ’s roots are approximately0 . , − . ± . √−
1, which also implies F cannot be normal). Therefore,the normal closure of F is a degree 6 extension K . The group U + K has rank 2 since K has no real embeddings, while a and one of its conjugates are multiplicativelyindependent units in U + K , neither of which lies on S . This means that U K ∩ S must be finite, so K and F are strictly hyperbolic.(5) Let us end with an example of a hyperbolic number field that is not strictlyhyperbolic. Let a, b, c be the roots of p ( x ) = x +3 x − a is the real root as inthe previous example. It is clear that | b | = 1 / √ a . Let E = Q ( √ a ), which is a real,degree 6 extension of Q and let E ′ = Q ( √ b ) and E ′′ = Q ( √ c ) be the conjugatesof E . It is claimed that E is hyperbolic, that is, if u ∈ U E has a conjugate v ofunit modulus, then u = ±
1. To verify this claim, let u ∈ E have a conjugate ofunit modulus. Without loss of generality, this conjugate can be assumed to besome v ∈ E ′ . Since ¯ b = c , one sees that ¯ v ∈ E ′′ and that v ¯ v = 1 implies that v, ¯ v ∈ E ′ ∩ E ′′ . The field E ′ ∩ E ′′ is of degree 1 , , E ′ = E ′′ , so its degree is 1 , E ′ ∩ E ′′ cannot be 3 so it mustbe 1 or 2. If the degree is 1, then the claim is proved; if the degree is 2, then v is a unit in a complex quadratic number field, hence v is a root of unity. This impliesthat u is a root of unity in the real field E , hence u = ± L of E contains √ a and b and thereforethe unit modulus number η = b √ a . If η were an n -th root of unity, then 1 = η n = b n a n ; but a and b are multiplicatively independent in K = Q ( a, b, c ), so n = 0.This shows that η ∈ U L ∩ S has infinite order and completes the proof that E ishyperbolic but not strictly hyperbolic.Let us turn to a theorem which demonstrates the importance of strictly hyper-bolic number fields. The choice of the set B F involves an arbitrariness which it hasbeen possible to avoid up to this point. To work around this arbitrariness, let themap πππ be extended to a map of G F by πππ ( τ ) = πππ (¯ τ ) ∀ τ B F . (6.20) Theorem 6.11. If F is strictly hyperbolic, then there is a β ∈ Aut( F/ Q ) such that (1) the induced maps U F / R F αβ / / U F / R F coincide; (2) πππ ( τ ) = τ ◦ β − ∀ τ ∈ G F ;(3) f = R β − (cid:12)(cid:12) V o,F where R β : V F → V F is the linear transformation inducedby precomposition with β ∈ Aut( F/ Q ) . Recall that R F is the set of units in U F all of whose conjugate lie on S . If F isstrictly hyperbolic, then R F = U F ∩ S . Proof.
For the purposes of this proof, it is convenient to extend α ∈ Aut( U + F ) to anautomorphism of U F = U + F ⊕ R F by extending α as the identity on R F . The choiceof extension of α is immaterial. The extension of α permits the extension of thehomomorphism ζ τ (Lemma 6.10), too. Since F is strictly hyperbolic, all conjugatesof U F ∩ S lie in S . Since α maps U F ∩ S to itself, this implies that the extendedhomomorphism ζ τ maps U F into S .Let U F = ∩ τ ∈ B F ker ζ τ . Since U K ∩ S is finite, ker ζ τ is a finite-index subgroupof U + F for all τ ; thus U F is a finite-index subgroup. Lemma 6.10.1 implies that ∀ u ∈ U F , τ ∈ B F : σ ( α ( u )) = τ ( u ) where πππ ( τ ) = σ. (6.21)This implies that σ ( U F ) ⊂ τ ( U F ); and since σ, τ are injective, the group σ ( U F ) is afinite-index subgroup of τ ( U F ). Therefore, τ ( F ) ∩ σ ( F ) contains elements that areof degree deg F . Thus, the two fields coincide: ∀ τ ∈ B F : τ ( F ) = σ ( F ) where πππ ( τ ) = σ. (6.22)Fix σ, τ ∈ G F with πππ ( τ ) = σ and define β σ := σ − ◦ τ. (6.23)Then, β σ | U F = α | U F and β σ ∈ Aut( F/ Q ). Because U F is a finite-index subgroupof U F it contains elements of degree deg F . It is clear that two automorphisms of F/ Q which coincide on an element of degree deg F , coincide on F . Therefore, thereis a single β ∈ Aut( F/ Q ) such that β σ = β for all σ .Moreover, from (6.22) and the remarks in the first paragraph, one knows that ζ σ maps U F into U F ∩ S . Consequently, σ − ◦ ζ σ maps U F into U F ∩ S . Since α ( u ) = σ − ( ζ σ ( u )) · β ( u ) ∀ u ∈ U F , (6.24)one sees that the invariance of S under embeddings of F implies that the inducedmaps U F / U F ∩ S αβ / / U F / U F ∩ S are equal. Since U + F is a non-canonical lift-ing of U F / U F ∩ S to U + F , one may declare that α = β | U + F . NTROPY AND TODA LATTICES 35
Finally, equation (6.23) implies that ∀ τ ∈ B F : πππ ( τ ) = σ ⇐⇒ σ = τ ◦ β − . (6.25)Therefore, the way in which πππ is extended to G F shows that πππ ( τ ) = τ ◦ β − for all τ ∈ G F . This proves the theorem. (cid:3) Topological conjugacy classes.
The results of the previous section affordthe opportunity to classify the Hamiltonian flows of the Bogoyavlenskij-Toda-typeHamiltonians (equation 3.16) up to topological conjugacy — at least in some situ-ations.
Standing Hypothesis : For the remainder of section 6, unless explicitly statedotherwise, it is assumed that F is a strictly hyperbolic number field.6.5.1. Root bases and Dynkin Diagrams.
Recall that for each root basis Ψ there is alabelled graph Γ(Ψ), called the Dynkin diagram, whose vertices are the points of Ψ.A pair of distinct vertices r, s have 4 hh r, s ii / | r | | s | edges connecting them, and if | r | > | s | then there is an arrow pointing from r to s . The vertex r has the label ω r .The Coxeter diagram is obtained from the Dynkin diagram by erasing the labels andarrows. If Ψ is a root system other than A (2)2 n , then one says that a permutation ρ ∈ S (Ψ) is an automorphism of the Dynkin diagram Γ(Ψ) iff the permutationleaves the Dynkin diagram unchanged with the exception of the numbering of theroots. Aut(Γ(Ψ)) is the automorphism group of Γ(Ψ). Note that ρ ∈ Aut(Γ(Ψ))iff ω r = ω ρ ( r ) and hh r, s ii = hh ρ ( r ) , ρ ( s ) ii for all r, s ∈ Ψ. For the root system A (2)2 n ,one defines the automorphism group, Aut(Γ( A (2)2 n )), to be the group generated bythe permutation that maps r j → r n +2 − j for all j (see figures 8–9).In the above discussion, one sees that the Cartan-Killing form must be nor-malised. We adopt the following normalisation: the shortest roots of D (2) n +1 and A (2)2 n have length 1 / √
2; all other root systems’ shortest roots have unit length.This normalisation implies that the longest root(s) of G (1)2 and D (3)4 have length √
3, while all other root systems’ longest roots have length √ Proposition 6.1.
Assume that F/ Q is strictly hyperbolic and B F > . Let ρ i ∈ B i = B ( B F , ΨΨΨ i ) be bijections and let H i be defined by Equation 3.16 withHamiltonian flow ϕ i . If there is an energy-preserving conjugacy of ϕ with ϕ ,then µ (equation 6.7) induces ν : ΨΨΨ → ΨΨΨ which is an isomorphism of Coxeterdiagrams. Thus, Case A: if ΨΨΨ n C (1) n , A (2)2 n , D (2) n +1 o , then (a) ΨΨΨ = ΨΨΨ ; (b) the constants c = c in the definition of Φ i (theorem 3.2); (c) ν ∈ Aut(Γ(Ψ)) . Case B: if ΨΨΨ ∈ n C (1) n , A (2)2 n , D (2) n +1 o , then (a) ΨΨΨ ∈ n C (1) n , A (2)2 n , D (2) n +1 o ; (b) the constants c and c in the definition of Φ i (theorem 3.2) are relatedby the following diagram: C (1) n × } } {{{{{{{{ × (cid:22) (cid:22) A (2)2 n × × / / D (2) n +1 × b b DDDDDDDD × u u where the factor ⋆ yields c = c × ⋆ . (c) if ΨΨΨ = ΨΨΨ , then ν ∈ Aut(Γ(Ψ)) .Proof.
From equation (6.15) and corollary 6.4, one knows that µ in equation (6.7)maps a root in h ∗ to a non-zero multiple of a root in h ∗ . Let ν denote the inducedbijection ΨΨΨ → ΨΨΨ . Since µ maps r ∈ Ψ to a scalar multiple of the root ν ( r ) ∈ Ψ ,one can write µ ( r ) = a r ν ( r ) for some coefficients a r .To determine the coefficients a r , note that φ i ( τ ) = n − τ ω r i r i where ρ i ( τ ) = r i .One computes that µ ( r ) = c c × φ ◦ f ∗ ◦ φ − ( r ) by definition of µ, = c c × n τ ω r × φ ◦ f ∗ (ˆ τ ) where ρ ( τ ) = r, = c c × n σ ω r × φ (ˆ σ ) where n σ ˆ σ = n τ f ∗ ˆ τ , = c c × ω ν ( r ) ω r × ν ( r ) where ρ ( σ ) = s, φ (ˆ σ ) = ω s n σ s and ν ( r ) = s. Therefore, since µ is an isometry hh r, s ii = (cid:18) c c (cid:19) × ω ν ( r ) ω ν ( s ) ω r ω s × hh ν ( r ) , ν ( s ) ii ∀ r, s ∈ Ψ . (6.26)This implies that the Coxeter diagrams of ΨΨΨ and ΨΨΨ are isomorphic. Inspectionof figures 8–9 shows that { ΨΨΨ , ΨΨΨ } is contained in one of the following sets: n A (1) n o n B (1) n , A (2)2 n − o n C (1) n , A (2)2 n , D (2) n +1 on D (1) n o n G (1)2 , D (3)4 on E (1) n o n =6 , , n F (1)4 , E (1)6 o (6.27)Case A. Suppose that we are in one of the cases covered by the first two columns of(6.27). Note that since c i ∈ N , one obtains that | r | | ν ( r ) | = c c × ω ν ( r ) ω r ∈ Q ∀ r ∈ Ψ . (6.28)The possible ratios of root lengths is 1 , √ √ ν isitself an isometry. Therefore ΨΨΨ = ΨΨΨ .To prove that c = c , note that since ν is a permutation of ΨΨΨ, it hasfinite order. If r ∈ Ψ is a fixed point of ν k for some k ≥
1, then equation6.28 implies that1 = (cid:18) c c (cid:19) k × ω ν ( r ) ω r × ω ν ( r ) ω ν ( r ) × · · · × ω ν k ( r ) ω ν k − ( r ) , (6.29)so 1 = (cid:16) c c (cid:17) k .Case B. In this case, following equation (6.28), the rational ratios that are possibleare 1 , /
2. A simple check shows that the natural Coxeter isomorphisms C (1) n → A (2)2 n , A (2)2 n → D (2) n +1 and D (2) n +1 → C (1) n satisfy this constraint with c /c equal to 1, 1 / = ΨΨΨ , then these considerations imply that ν is a Dynkin diagramautomorphism and c = c . (cid:3) NTROPY AND TODA LATTICES 37 C (1) n A (2)2 n D (2) n +1 ✻✻ ✻✻ ✻✻ ✻✻◆ ✌◆✌ Figure 3.
The natural Coxeter isomorphisms ν : ΨΨΨ → ΨΨΨ . Remark 6.3.
In [11, Lemma 24] there is a simpler version of theorem 6.1. It isassumed there that ΨΨΨ i = A (2)2 n for both i and c = c . In this case, ν must be anautomorphism of the Dynkin diagram.7. Topological Entropy
In the proof of the complete integrability, Theorem 4.1, one sees that the singularset of the algebra L + R is the union of L − ( R c ) and Z = k − (0). Theorem 5.1shows that the non-wandering set of the Hamiltonian flow ϕ of H , restricted to theinvariant set Z = W ± ( V ⊥ ), is V ⊥ . What happens on the other part of the singularset, L − ( R c )?7.1. The A (1) n lattice. Thanks to the work of Foxman and Robbins [15, 16], thisquestion is answerable for the A (1) n lattice. Theorem 7.1.
Let
ΨΨΨ = A (1) n and H be a Bogoyavlenskij-Toda-like Hamiltoniandefined in equation 3.16. Then L − ( R c ) is stratified by symplectic manifolds thatare invariant under the Hamiltonian flow of H . Moreover, H restricted to eachstratum is completely integrable.Proof. Let h = 2 κ ∈ C ∞ (e+ L ∗ + L ∗ +1 ) be the Cartan-Killing form, so that H = L ∗ h (equation 3.16). Foxman and Robbins [15, 16] proved that h admits action-anglevariables with singularities, which means that for each point p ∈ e+ L ∗ + L ∗ +1 , thereare coordinates ( x, y, u, v ) on a neighbourhood of p in O Rp such that the canonicalsymplectic form on O p and h take the form k X i =1 d x i ∧ d y i + n X i = k +1 d u i ∧ d v i , h = h ( x, ρ ) ρ i = u i + v i , i = k + 1 , . . . n, where k = 0 , . . . , n is the co-rank of the singularity (and if k = n , then there isno singularity). The set ρ = 0 is an invariant symplectic submanifold and h iscompletely integrable on this submanifold.Let X k ⊂ e + L ∗ + L ∗ +1 be one of these symplectic sub-manifolds of dimension2 k and co-dimension 2 l where n = k + l . Let Y k = L − ( X k ).Because L | k − ( R −
0) is a submersion onto e + L ∗ + L ∗ +1 , Y k is a submanifoldof T ∗ Σ of co-dimension 2 l . Moreover, since L is a Poisson submersion, Y k is also asymplectic submanifold. Since X k is invariant under the hamiltonian flow of h , Y k is similarly invariant.From the above description of the singular action-angle variables, the algebra L | Y k is equal to L ∗ Z ∞ ( L ∗ ) | X k , which contains k functionally independent elements.On the other hand, the algebra R | Y k contains dim V E functionally independent ele-ments. Therefore, in total, there are k + dim V E functionally independent integrals of H at each point of Y k . Since dim Y k = 2( k + dim V E ), this proves the completeintegrability of H | Y k , which proves the theorem. (cid:3) Remark 7.1.
It is clear from the proof that H | U is completely integrable withsingular action-angle variables. This is the mildest kind of singularity that a com-pletely integrable may have. It is a stark contrast with the sort of singularity thatdevelops along Z = k − (0).It is natural to conjecture that the Foxman-Robbins theorem is true for allBogoyavlenskij-Toda lattices. Corollary 7.2.
Let
ΨΨΨ = A (1) n and H be a Toda-like hamiltonian defined in equation3.16. The topological entropy of ϕ | H − ( ) , the time- map of the hamiltonian flowof H , equals h top = [ E : F ] c × s floor (cid:18) n + 12 (cid:19) (7.1) where n = dim V o,F and c ∈ Z + as in theorem 3.2.Proof. Since ϕ admits singular action-angle variables on k − ( R − ϕ is generated entirely in k − (0) = W ± ( V ⊥ ). The non-wandering set of ϕ | W ± ( V ⊥ ) is V ⊥ by theorem 5.1. Thus h top ( ϕ | H − ( 12 )) = h top ( ϕ | V ⊥ ) = [ E : F ] c × s floor (cid:18) n + 12 (cid:19) by table 5. (7.2) (cid:3) The remaining Bogoyavlenskij-Toda lattices.
As in [11, Section 3], theuniversal covering space ˜Σ = V E × V o,F admits the structure of a solvable Lie group.The element v ∈ V o,F acts by right translation by the one-parameter subgroup˜ φ vt ( y , x ) = ( y + t · v, x ) . (7.3)This flow descends to a flow φ v on Σ. As in [11, Lemma 12], h top ( φ v ) = [ E : F ] × X τ ∈ B F n τ h ˆ τ , v i + (7.4)where u + = max { u, } .Let H be defined by equation (3.16) and let V ⊥ = V ⊥ ∩ H − ( 12 ) (7.5)where V ⊥ is defined in section 5. If v = Q · X with X ∈ V ∗ o,F , and hQ · X , X i = 1,then ∆ · ( y , x , , X ) ∈ V ⊥ . The topological entropy of the Hamiltonian flow ϕ of H is therefore equal to1[ E : F ] × h top ( ϕ | V ⊥ )= max X : hQ· X , X i =1 X τ ∈ B F n τ h ˆ τ , Q · X i + = max X : hQ· X , X i =1 X τ ∈ B F n τ hh φ ρ (ˆ τ ) , φ ρ ( X ) ii = max s ∈ h : hh s,s ii =1 X r ∈ ΨΨΨ ω r c × h r, s i + where φ ρ (ˆ τ ) = ω r n τ c r, s = φ ρ ( X )= c − × max I ⊂ ΨΨΨ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X r ∈ I ω r r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7.6) NTROPY AND TODA LATTICES 39
The right-hand side of 7.6 is computed in [11, Lemma 13 and Theorem 3]. Theseresults are summarised in table 5. h top ( ϕ | V ⊥ ) = h × [ E : F ] c ΨΨΨ h ΨΨΨ h ΨΨΨ hB (1) n , n ≥ √ n − A (2)2 n − , n ≥ p n − G (1)2 , ( n = 2) 2 √ D (3)4 , ( n = 2) 2 F (1)4 , ( n = 4) 2 √ E (2)6 , ( n = 4) 2 √ C (1) n , n ≥ √ n A (2)2 n , n ≥ √ n D (2) n +1 , n ≥ √ nE (1)6 , ( n = 6) 2 √ E (1)7 , ( n = 7) 2 √ E (1)8 , ( n = 8) 2 √ A (1) n , n ≥ q floor (cid:0) n +12 (cid:1) D (1) n , n ≥ p n − A (1)2 , ( n = 1) √ Table 5.
Entropies of the Bogoyavlenskij-Toda-like systems. Theroot systems in the first 4 rows have isomorphic Coxeter graphs;the root systems in the last 2 rows have unique Coxeter graphs. n = dim V o,F .Table 5 permits one to give lower bounds on the number of Bogoyavlenskij-Toda-like systems which are not energy-preserving topologically conjugate. Proposition 7.1.
For each n ≥ , table 6 displays Bogoyavlenskij-Toda-like sys-tems, defined in (3.16), that are not topologically conjugate via an energy-preservingconjugacy. n RootSystems Total2 A (1)2 , C (1)2 , G (1)2 , A (2)2 · A (1)3 , C (1)3 , A (2)2 · , A (2)2 · − A (1)4 , B (1)4 , A (2)2 · , A (2)2 · − A (1)5 , B (1)5 , C (1)5 , D (1)5 , A (2)2 · , A (2)2 · − A (1)6 , B (1)6 , D (1)6 , A (2)2 · , A (2)2 · − A (1)7 , B (1)7 , C (1)7 , D (1)7 , A (2)2 · , A (2)2 · − A (1)8 , B (1)8 , C (1)8 , D (1)8 , E (1)8 , A (2)2 · − ≥ A (1) n , B (1) n , D (1) n , A (2)2 · n , A (2)2 · n − ≥ A (1) n , B (1) n , C (1) n , D (1) n , A (2)2 · n , A (2)2 · n − Table 6.
Minimal number of Bogoyavlenskij-Toda-like systemsthat are not iso-energetically topologically conjugate.
Proof.
Use table 5 to determine a list of root systems the ratio of whose entropiesdo not lie in Z . Note that this list is not unique. (cid:3) Summary.
If the results from table 5 are combined with proposition 6.1, oneobtains the much stronger result:
Theorem 7.3.
Let F/ Q be a strictly hyperbolic number field with n + 1 = B F > . The number of iso-energetic topological conjugacy classes of Hamiltonian flowsconstructed from Equation (3.16) is at least X rankΨΨΨ= n \ B (ΨΨΨ) / Aut( F/ Q )) . (7.7) where we sum over all rank n root systems except D (2) n +1 .Proof. By Proposition 6.1, we know that if the Bogoyavlenskij-Toda-like Hamilton-ian flows ϕ i are conjugate by an energy-preserving conjugacy, then there are twopossibilitiesCase A. The root systems coincide, c = c and the map ν = µ is an automorphismof Γ(ΨΨΨ). The definition of µ (equation 6.7 and supra φ , φ are related by φ = µ · φ · R ∗ β β ∈ Aut( F/ Q ) (7.8)where theorem 6.11 is used. Conversely, given any φ , a φ defined as inequation (7.8) is induced by a bijection ρ ∈ B .Case B. The two root systems differ, as in Case B of Proposition 6.1. The topologicalentropy of ϕ i | V ⊥ ∩ H i ( ) is an invariant of energy-preserving conjugacyby Lemma 6.2. Table 5 implies that the root systems must therefore be { ΨΨΨ , ΨΨΨ } = n A (2)2 n , D (2) n +1 o or n C (1) n , D (2) n +1 o . Since the sum (7.7) countsthe conjugacy classes from only one of these two root systems, there is nodouble counting. This proves the theorem. (cid:3) Remark 7.2.
In [11, Example 3, p. 541], the case where F = E = Q ( α ), with α a root of the cubic x − x + 2, was considered ( c.f. example 4.1.1 supra ). F is acubic, totally-real, non-normal extension of Q . Thus, Aut( F/ Q ) is trivial and F isstrictly hyperbolic. If one sums over the rank 2 root systems and divides out bythe order of their automorphism groups, then Theorem 7.3 implies that there areat least1 + 3 + 6 + 3 + 6 = 19 (summing over A (1)2 , C (1)2 , G (1)2 , A (2)2 · , D (3)4 ) (7.9)iso-energetic topological conjugacy classes. In [11, theorem 8], the lower bound of 10was conjectured. This lower bound depended on Gel’fond’s conjecture concerningthe algebraic independence of rationally-independent sets of logarithms of algebraicnumbers. The results of the present paper, using dynamical systems theory, hasproven this lower bound.In a similar vein, if F = E is a totally real quartic field with Aut( F/ Q ) = 1,then one has at least3 + 4 ×
12 = 51 (summing over A (1)3 , B (1)3 , C (1)3 , A (2)2 · , A (2)2 · − ) (7.10)iso-energetic topological conjugacy classes. Remark 7.3.
Theorem 7.3 provides a means to compute a lower bound on thenumber of iso-energetic topological conjugacy classes when Aut( F/ Q ) is non-trivial,too. Both ΨΨΨ and B F are unnaturally isomorphic to the set { , . . . , n + 1 } . Theorem6.11, part 2, shows that the representation of Aut( F/ Q ) in the group of permuta-tions of B F , S ( B F ), is the natural right regular representation (one should view B F = G F / ( · ∼ ¯ · )). By definition, the automorphism group of the Dynkin diagramis a subgroup of the group of permutations of the roots, S (ΨΨΨ). Therefore, the Inexplicably, only the first three root systems are included in that sum, so the conjecturallower bound ought to be 19.
NTROPY AND TODA LATTICES 41 unnatural isomorphisms of ΨΨΨ and B F with { , . . . , n + 1 } identify the set of bijec-tions B (ΨΨΨ) with the symmetric group of { , . . . , n + 1 } , S n +1 , with the resultingequivariant diagram (where left/right arrows denote the standard left (resp. right)actions)Aut(Γ(ΨΨΨ)) (cid:31) (cid:127) / / s(cid:19) % % KKKKKKKKKK ∼ = (cid:15) (cid:15) S (ΨΨΨ) / / ∼ = (cid:15) (cid:15) B (ΨΨΨ) ∼ = (cid:15) (cid:15) S ( B F ) o o ∼ = (cid:15) (cid:15) Aut( F/ Q ) ? _ o o kK x x rrrrrrrrrr ∼ = (cid:15) (cid:15) G (cid:31) (cid:127) / / S n +1 id . / / S n +1 S n +1id . o o H. ? _ o o (7.11)This implies that G \ S n +1 /H ) equals \ B (ΨΨΨ) / Aut( F/ Q )). Table 7shows the cardinality of each of these sets for n ≤
9. The table is computed bya C++ program written by the author; the computations were checked using theGAP software package [18]. The source code and instructions are freely availablefrom the author’s web-page.
Table 7.
The minimum number of iso-energetic topological con-jugacy classes of Bogoyavlenskij-Toda-like systems. The
Total col-umn is based on Theorem 7.3 and Tables 8–9 of Coxeter graphautomorphism groups. Z n = Z /n Z , D n = the dihedral group of order 2 n , Q = the quaterniongroup of order 8. \ B (ΨΨΨ) / Aut( F/ Q )) . rankGalois grp Root systems (grouped with isomorphic Coxeter diagrams) Totalrank = 2Aut( F/ Q ) A (1)2 C (1)2 /A (2)2 · /D (2)2+1 G (1)2 /D (3)4 Total1 1 3 × × Z × × F/ Q ) A (1)3 C (1)3 /A (2)2 · /D (2)3+1 B (1)3 /A (2)2 · − Total1 3 12 × × Z ⊕ Z × × Z × × F/ Q ) A (1)4 C (1)4 /A (2)2 · /D (2)4+1 B (1)4 /A (2)2 · − D (1)4 F (1)4 /E (2)6 Total1 12 60 × × × Z × × × F/ Q ) A (1)6 C (1)6 /A (2)2 · /D (2)6+1 B (1)6 /A (2)2 · − D (1)6 Total1 60 360 × × Z
14 64 × × S
19 72 × × F/ Q ) A (1)6 C (1)6 /A (2)2 · /D (2)6+1 B (1)6 /A (2)2 · − D (1)6 E (1)6 Total1 360 2 520 × × Z
54 360 × × F/ Q ) A (1)7 C (1)7 /A (2)2 · /D (2)7+1 B (1)7 /A (2)2 · − D (1)7 E (1)7 Total continued next page
Table 7, continued from previous page × × Z
332 2 544 × × Z
420 2 688 × × Z ⊕ Z
362 2 592 × × Q
333 2 544 × × D
391 2 640 × × F/ Q ) A (1)8 C (1)8 /A (2)2 · /D (2)8+1 B (1)8 /A (2)2 · − D (1)8 E (1)8 Total1 20 160 181 440 × × Z × × Z × × F/ Q ) A (1)9 C (1)9 /A (2)2 · /D (2)9+1 B (1)9 /A (2)2 · − D (1)9 Total1 181 440 1 814 400 × × Z
18 264 181 632 × × D
18 724 182 400 × × Conclusion
The current paper shows that there is a rich family of completely integrableHamiltonian systems to be found on the cotangent bundles of compact 2-step
Sol -manifolds. In addition to the questions in the introduction, let us mention thefollowing question which arises from lemma 6.10 and theorem 6.11.
Question F.
Let F be a number field that is not strictly hyperbolic. Assume thatthere is an automorphism α of U F and a permutation πππ of G F such that ∀ u ∈ U F , ∀ τ ∈ G F : | τ ( α ( u )) | = | σ ( u ) | where πππ ( τ ) = σ, and (8.1) ∀ τ ∈ G F : πππ (¯ τ ) = πππ ( τ ) Is it true that there is an automorphism β of F/ Q such that α = β | U F ? In otherwords, is it true that ∩ τ ∈ G F ker ζ τ is always a finite-index subgroup of U F ? It appears the likely that the answer is yes . To explain: If u i is a basis of U + F and α ∈ Aut( U F ), then α ( u i ) = ǫ i × Q j u a ji j for some integer matrix A = [ a ji ] thatis invertible over the integers, and some root of unity ǫ i ∈ U F . From the condition(8.1), one knows that the system of linear equations X j a ji ln | πππσ ( u i ) | = ln | σ ( u i ) | (8.2)is satisfied for all j = 1 , . . . , B F − σ ∈ B F . For a fixedpermutation πππ , one can treat (8.2) as a linear system that determines A . If thereis an integer solution, then this determines an automorphism α ; if not, then thereis no such automorphism.Salem number fields are good candidates to investigate question F becausethese number fields have many infinite order units of modulus one. By meansof Maxima [24], it has been numerically verified that the answer to the refined ques-tion is yes for the 13 lowest degree number fields generated by the ‘small’ Salemnumbers listed by Mossinghoff, based on [8, Table 1] and [25, Table 1].
NTROPY AND TODA LATTICES 43
RootSystem Dynkin Diagram root numberweight AutomorphismGroup A (1)1
11 21 Z A (1) n
11 21 n − n n + 11 D n +1 ( n ≥ B (1) n n + 1111 22 n − n Z ( n ≥ C (1) n n + 11 12 n − n Z ( n ≥ D (1) n n + 1111 22 n − n − n S ( n = 4) Z ( n > G (1)2
23 12 31 1 F (1)4
51 12 23 34 42 1 E (1)6
11 32 4 3 52 612 27 1 D E (1)7
81 12 33 4 4 53 62 712 2 Z E (1)8
12 34 4 6 55 64 73 82 912 3 1
Table 8.
Root systems, their Dynkin diagrams and automor-phism groups. Symmetries are indicated by arrows. D n is thesymmetry group of a regular n -gon. RootSystem Dynkin Diagram root numberweight AutomorphismGroup A (2)2
11 22 1 A (2)2 n
12 22 n n + 11 Z (see text, n ≥ A (2)2 n − n + 1111 22 n − n Z ( n ≥ D (2) n +1
11 21 n n + 11 Z ( n ≥ E (2)6
51 12 23 32 41 1 D (3)4
11 22 31 1
Table 9.
Root systems, their Dynkin diagrams and automor-phism groups. The shortest roots of D (2) n +1 and A (2)2 n have length1 / √
2; all other root systems’ shortest roots have unit length. Thelongest root(s) of G (1)2 and D (3)4 have length √
3; all other rootsystems’ longest roots have length √ NTROPY AND TODA LATTICES 45
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