A short treatise on the equivariant degr

A short treatise on the equivariant degree theory and its applications Journal of Fixed Point Theory and Applications ISSN 1661-7738 Volume Number J Fixed Point Theory Appl (2010) 8:1-74 DOI 10.1007/ s11784-010-0033-9 23 Your article is protected by copyright and all rights are held exclusively by Springer Basel AG This e-offprint is for personal use only and shall not be self-archived in electronic repositories If you wish to self-archive your work, please use the accepted author’s version for posting to your own website or your institution’s repository You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication 23 Author's personal copy J Fixed Point Theory Appl (2010) 1–74 DOI 10.1007/s11784-010-0033-9 Published online October 27, 2010 © Springer Basel AG 2010 Journal of Fixed Point Theory and Applications A short treatise on the equivariant degree theory and its applications Zalman Balanov, Wieslaw Krawcewicz, Slawomir Rybicki and Heinrich Steinlein Dedicated to Stephen Smale Abstract The aim of this survey is to give a profound introduction to equivariant degree theory, avoiding as far as possible technical details and highly theoretical background We describe the equivariant degree and its relation to the Brouwer degree for several classes of symmetry groups, including also the equivariant gradient degree, and particularly emphasizing the algebraic, analytical, and topological tools for its effective calculation, the latter being illustrated by six concrete examples The paper concludes with a brief sketch of the construction and interpretation of the equivariant degree Mathematics Subject Classification (2010) Primary 47H11; Secondary 34C60, 37G40, 70F10, 92C15 Keywords Equivariant (gradient) degree, Burnside ring, multiplicativity, recurrence formula, basic degree, Euler ring, equivariant Hopf bifurcation, legged locomotion, Lennard–Jones 3-body problem, delay system of variational type, Newtonian system The author of this book considers this work to be the shorter the better Who finds this book to be too long now, can be sure that it will become shorter and shorter with every further edition until it comes close to the ideal: “It cannot be compared with silence” Sholom Aleichem “Menachem Mendel” Author's personal copy Z Balanov et al JFPTA Introduction 1.1 Subject The equivariant degree is a topological tool allowing “counting” orbits of solutions to (symmetric) equations in the same way as the usual Brouwer degree does it, but according to their symmetric properties This method is an alternative and/or complement to the equivariant singularity theory developed by M Golubitsky et al (see, for example, [53]), as well as to a variety of methods rooted in Morse theory, Lusternik–Schnirelman theory and Morse–Floer complex techniques (see, for example, [14, 15, 78, 87]) used for a treatment of variational problems with symmetries These standard methods, although being quite effective in the settings they are usually applied, encounter technical difficulties when (i) the group of symmetries is large, (ii) multiplicities of eigenvalues of linearizations are large, (iii) phase spaces are of high dimension, (iv) operators involved exhibit lack of smoothness, respectively Also, one would expect to use computer routines for complex computations, while it is not clear if these approaches are “open enough” to be computerized On the other hand, the equivariant degree theory has all the attributes allowing its application in settings related to (i)–(iv) and, in many cases, allows a computerization 1.2 Goal The equivariant degree theory, which emerged in nonlinear analysis about 20 years ago, has many natural roots: Borsuk–Ulam theorems (see [98, 72, 60]), fundamental domains and equivariant retract theory (see [72, 2, 13, 60]), equivariant homotopy groups of spheres (see [73, 33, 60, 13, 75]), equivariant bordism theory (see [99, 13]), equivariant general position theorems (see [72, 46, 60]), gradient S -invariants (see [28, 31]), Whitehead J-homomorphism (see [56]) Two monographs on this subject (see [13, 60]) can provide an experienced reader with a systematic exposition of the equivariant degree theory in all its aspects (see also the survey papers [57, 7, 96]) However, as a matter of fact, the equivariant degree theory is a combination of algebraic, topological and analytical methods, and only in this capacity is effective when applied to study symmetric equations Unfortunately, its theoretical sophistication may be too advanced for applied mathematicians to whom this is first and foremost addressed Nevertheless, clear properties, rules of usage and computer assisted routines provide easy steps for effective applications of the equivariant degree in various applied areas The goal of this paper is to show that the equivariant degree can be effectively used outside its theoretical context and foundations 1.3 Overview After the Introduction, the paper is organized as follows In Section 2, together with the standard preliminaries, we discuss symmetries of periodic functions which are crucial for understanding the main idea behind the applicability of the equivariant degree to study periodic solutions to dynamical Author's personal copy Vol (2010) Equivariant degree theory systems In fact, the equivariant degree has different faces reflecting a diversity of symmetric equations related to applications Therefore, in Sections 3–5, we discuss in a parallel way three variants of the equivariant degree In Section 3, we consider equivariant maps without free parameter: starting with axioms of the Brouwer degree, we show how they should be modified to describe the corresponding equivariant counterpart In Section 4, we consider equivariant maps with one free parameter: starting with the axioms of the S -degree, we outline an axiomatic approach to the twisted equivariant degree In Section 5, starting with a short discussion on the Hopf property for G-gradient maps, we present the concept of the G-gradient degree Section is devoted to applications: each variant of the equivariant degree is illustrated by corresponding examples (symmetric BVP for the equivariant degree without parameter, symmetric Hopf bifurcation (local and global) for the twisted degree, relative equilibria of the Lennard–Jones system, symmetric delay system of variational type and symmetric Newtonian system for the G-gradient degree) In Section 7, we briefly discuss the existence of the equivariant degree, i.e., constructions, extensions to the infinite dimensional case, etc In particular, several parts of the general equivariant degree which are not mentioned in Sections 3–5, (for example, the so-called secondary degree) are discussed here Preliminaries 2.1 Basic definitions Throughout this paper, G stands for a compact Lie group For a subgroup H ⊂ G (which is always assumed to be closed), denote by N (H) the normalizer of H in G, by W (H) = N (H)/H the Weyl group of H in G, and by (H) the conjugacy class of H in G The set Φ(G) of all conjugacy classes in G admits a partial order defined as follows: (H) ≤ (K) iff gHg −1 ⊂ K for some g ∈ G In addition, Φ(G) can be “stratified” by the subsets Φn (G) := {(H) ∈ Φ(G) : dim W (H) = n} For a G-space X and x ∈ X, denote by Gx := {g ∈ G : gx = x} the isotropy of x, by G(x) := {gx : g ∈ G} G/Gx the orbit of x and by X/G the orbit space Given an isotropy Gx , call (Gx ) the orbit type in X and put Φ(G; X) := {(H) ∈ Φ(G) : H = Gx for some x ∈ X}, Φn (G; X) := Φn (G) ∩ Φ(G; X) Also, adopt the following notations: XH := {x ∈ X : Gx = H}, X H := {x ∈ X : Gx ⊃ H}, X(H) := {x ∈ X : (Gx ) = (H)}, X (H) := {x ∈ X : (Gx ) ≥ (H)} As is well known (see, for instance, [33]), W (H) acts on X H and this action is free on XH Consider two subgroups L ⊂ H of G and put N (L, H) := {g ∈ G : gLg −1 ⊂ H} Then, N (L, H)/H is a (right) W (L)-space It is well known that (G/H)L (as a left W (L)-space) is W (L)-equivariantly diffeomorphic to N (L, H)/H and (G/H)L contains finitely many W (L)-orbits Also, N (L, H) is a left N (H)-space and, moreover (see [13, Proposition 2.52]), if dim W (L) = dim W (H), then n(L, H) := |N (L, H)/N (H)| is finite Author's personal copy Z Balanov et al JFPTA Let X and Y be two G-spaces A continuous map f : X → Y is said to be equivariant if f (gx) = gf (x) for all x ∈ X and g ∈ G If the G-action on Y is trivial, then f is called invariant Clearly, for any subgroup H ⊂ G and equivariant map f : X → Y , the map f H : X H → Y H , with f H := f |X H , is well defined and W (H)-equivariant Finally, given two Banach spaces E1 and E2 and an open bounded subset Ω ⊂ E1 , a continuous map f : E1 → E2 is called Ω-admissible if f (x) = for all x ∈ ∂Ω (in such a case we will also say that (f, Ω) is an admissible pair) Similarly, two Ω-admissible maps f and g are called Ω-admissible homotopic if there exists a continuous map h : [0, 1] × E1 → E2 (called Ω-admissible homotopy between f and g) such that (i) h(0, x) = f (x), h(1, x) = g(x) for all x ∈ E1 , and (ii) the map ht (x) := h(t, x) is Ω-admissible for all t ∈ [0, 1] Denote by M(E1 , E2 ) the set of all admissible pairs (f, Ω) For more information on the equivariant topology background used in this paper, we refer the reader to [33, 62, 20, 13] 2.2 Isotypical decomposition of finite-dimensional representations As is well known, any compact group admits only countably many nonequivalent real (resp., complex) irreducible representations Therefore, given a compact Lie group Γ, we always assume that a complete list of all real (resp., complex) irreducible Γ-representations, denoted Vi , i = 0, 1, (resp., Uj , j = 0, 1, ), is available Let V (resp., U ) be a finite-dimensional real (resp., complex) Γ-representation (without loss of generality, V (resp., U ) can be assumed to be orthogonal (resp., unitary)) Then, it is possible to represent V (resp., U ) as the direct sum V = V0 ⊕ V ⊕ · · · ⊕ V r , (2.1) (resp., U = U0 ⊕ U1 ⊕ · · · ⊕ Us ), (2.2) called the Γ-isotypical decomposition of V (resp., U ), where the isotypical component Vi (resp., Uj ) is modelled on the irreducible Γ-representation Vi , i = 0, 1, , r, (resp., Uj , j = 0, 1, , s); i.e., Vi (resp., Uj ) contains all the irreducible subrepresentations of V (resp., U ) which are equivalent to Vi (resp., Uj ) Given an orthogonal Γ-representation V , denote by LΓ (V ) (resp., Γ GL (V )) the R-algebra (resp., group) of all Γ-equivariant linear (resp., invertible) operators on V Then, the isotypical decomposition (2.1) induces a decomposition of GLΓ (V ): r GLΓ (V ) = GLΓ (Vi ), (2.3) i=0 and for every isotypical component Vi from (2.1), GLΓ (Vi ) GL(mi , F), where mi = dim Vi / dim Vi and F is a finite-dimensional division algebra LΓ (Vi ), i.e., F = R, C or H, depending on the type of the irreducible representation Vi Author's personal copy Vol (2010) Equivariant degree theory Since in the case F = C or H, the group GL(mi , F) is connected, for every A ∈ GLΓ (Vi ), there exists a continuous path τi : [0, 1] → GLΓ (Vi ) such that τi (0) = A and τi (1) = Id Vi On the other hand, if F = R, then GLΓ (Vi ) has two connected components: one of them contains Id Vi , while the other GL(mi , R)) contains the operator one (under the identification GLΓ (Vi ) diag {−1, 1, 1, , 1} 2.3 Symmetries of periodic functions Let V be a Euclidean space and consider the space E := C2π (R; V ) of all continuous 2π-periodic functions x : R → V The space E, equipped with the usual sup-norm · ∞ , is a Banach space Notice that, by using the identification S := R/2πZ (via the map τ → eiτ ), we can identify E with C(S ; V ), which is an isometric Banach S -representation with the S -action given by (2.4) (eiτ x)(t) = x(t + τ ), eiτ ∈ S and x ∈ E Remark 2.1 For G = S and x ∈ E, x is a constant function iff Gx = S , while x is a nonconstant periodic function iff Gx = Zk for some k ∈ N Since S is abelian, the conjugacy class (Zk ) consists of exactly the group Zk So far, only the symmetries induced by time shift have been involved To see an impact of spacious symmetries, assume that V is an orthogonal Γ-representation (with Γ finite, for simplicity) One can extend the S -action given by (2.4) to the Γ × S -action given by ((γ, eiτ )x)(t) = γx(t + τ ), (γ, eiτ ) ∈ Γ × S and x ∈ E (2.5) Clearly, E becomes an isometric Banach Γ × S -representation and we arrive at the following question, given x ∈ E, how the symmetries of x look like? Put G := Γ×S and assume x is a constant function Then, x is independent of time, hence S ⊂ Gx from which it follows that Gx = K ×S for some subgroup K ⊂ Γ, i.e., Gx is a product subgroup In addition, dim W (Gx ) = Moreover, Φ0 (G) is composed of conjugacy classes of product subgroups If x is not constant, then Gx must be a twisted subgroup, i.e., there exists a subgroup K ⊂ Γ, an integer l ≥ and a homomorphism ϕ : K → S such that (2.6) Gx = K ϕ,l := {(γ, z) ∈ K × S : ϕ(γ) = z l } If l > 1, then we also say that the twisted subgroup K ϕ,l is l-folded One can easily show that for a twisted subgroup K ϕ,l , dim W (K ϕ,l ) = Moreover, the set Φ1 (G) is composed of the conjugacy classes of twisted subgroups We summarize our observations in the following Remark 2.2 There is a parallelism between the analytic (constant/nonconstant functions), algebraic (product/twisted symmetries) and topological (zero/one-dimensional Weyl group) dichotomies As we will see later on, the equivariant degree is, to some extent, a tool allowing to “exploit” this parallelism in the realm of stationary/nonstationary periodic solutions to dynamical systems with symmetries Author's personal copy Z Balanov et al JFPTA From Brouwer degree to equivariant degree without free parameter 3.1 Axioms of Brouwer degree The Brouwer degree is a fundamental tool of nonlinear analysis The main idea behind its usage can be traced back to the intermediate value theorem (IVT) from the elementary calculus: given a continuous map f : [a, b] → R, the knowledge of the behavior of f only on the boundary of the segment (change of sign) implies the solubility of the equation f (x) = inside the segment Although the Brouwer degree is a simple integer-valued characteristic of an admissible pair, its careful definition based on elementary analysis or algebraic topology requires complicated constructions However, in applications it is enough to know that this characteristic exists and satisfies several fundamental properties (in particular, generalizing IVT to higher dimensions), which is possible to combine with techniques specific for a concrete nonlinear analytic setting This gives rise to the axiomatic approach to the Brouwer degree outlined below Put M := V M(V, V ), where V is a Euclidean space (see Subsection 2.1) The following theorem states the existence of the (local) Brouwer degree Theorem 3.1 There exists a unique map deg : M → Z which assigns to every admissible pair (f, Ω) an integer deg(f, Ω) ∈ Z satisfying the following properties: (B1) (Existence) If deg (f, Ω) = 0, then Ω contains a solution to the equation f (x) = (3.1) (B2) (Additivity) Let Ω1 and Ω2 be two disjoint open subsets of Ω such that f −1 (0) ∩ Ω ⊂ Ω1 ∪ Ω2 Then, deg(f, Ω) = deg(f, Ω1 ) + deg(f, Ω2 ) (B3) (Homotopy) If h : [0, 1] × V → V is an Ω-admissible homotopy, then deg(ht , Ω) = constant (B4) (Normalization) Let x0 ∈ Ω and f (x) = x − x0 Then, deg(f, Ω) = deg(Id − x0 , Ω) = Remark 3.2 Property (B3) states that the Brouwer degree is stable not only under small continuous perturbations, but also under large continuous deformations that allows to deform complicated equations to relatively simple ones Property (B2) allows to localize zeros of maps that provides a passage to the computation of the degree on smaller domains, while property (B4) indicates the simplest map with nonzero degree Property (B1) can be viewed as an n-dimensional analogue of the IVT Observe also that property (B1) follows from (B2)–(B4) meaning that the “axioms” (B2)–(B4) completely determine “deg” as a function of the two variables Ω and f Using (B1)–(B4) one can establish many other useful properties of the Brouwer degree Among them, the following two are of great importance (below W stands for another Euclidean space) Author's personal copy Vol (2010) Equivariant degree theory (B5) (Multiplicativity) For any (f1 , Ω1 ), (f2 , Ω2 ) ∈ M, deg(f1 × f2 , Ω1 × Ω2 ) = deg(f1 , Ω1 ) · deg(f2 , Ω2 ), where “·” stands for the usual multiplication in the ring Z (B6) (Suspension) If B is an open bounded neighborhood of ∈ W , then deg(f × Id W , Ω × B) = deg(f, Ω) Remark 3.3 The “functorial” property (B5) can be used to decompose a complicated map (in particular, to decrease dimension) into a product of two simpler maps, while property (B6) provides a passage to infinite dimensions which is usually the case in the applications of the Brouwer (more precisely, Leray–Schauder) degree to dynamical systems via the functional analysis approach The Brouwer degree is commonly known as an “algebraic count of solutions to equations.” However, property (B1) provides the existence of at least one solution with no multiplicity information Combining properties (B2)–(B4) with elementary notions of transversality theory, one can easily establish (B7) (Regular Value Property) Let (f, Ω) ∈ M, f is C -smooth and is a regular value of f Then, deg(f, Ω) = sign det Df (x) (3.2) x∈f −1 (0)∩Ω To complete this subsection, observe that given (f1 , Ω), (f2 , Ω) ∈ M(V, V ), from deg(f1 , Ω) = deg(f2 , Ω) it does not follow that f1 and f2 are Ω-admissible homotopic, in general Nevertheless, the following statement is true (B8) (Hopf Property) Assume B(V ) is the unit ball in V and for (f1 , B(V )), (f2 , B(V )) ∈ M, one has deg(f1 , B(V )) = deg(f2 , B(V )) Then, f1 and f2 are B(V )-admissible homotopic 3.2 The set MΓ and Burnside ring The Brouwer degree “deg” is a function defined on the set M of admissible pairs (occuring in Euclidean spaces) and taking its values in the ring Z Let Γ be a compact Lie group Then, a passage to the Γ-symmetric setting will imply appropriate replacements of M, Z (described in the present subsection) and deg (described in the remaining subsections) Recall that in this section we will be dealing with the simplest version of the equivariant degree (i.e., the one without free parameters) Thus, assume that V is equipped with the structure of an orthogonal Γ-representation and call (f, Ω) ∈ M(V, V ) an admissible Γ-pair if Ω is Γ-invariant and f is Γ-equivariant Denote by MΓ (V, V ) the set of all admissible Γ-pairs (f, Ω) ∈ M(V, V ) and put MΓ := V MΓ (V, V ), where V is an orthogonal Γ-representation (this family is a replacement of M) In an obvious way Author's personal copy Z Balanov et al JFPTA one can define an Ω-admissible Γ-homotopy Next, the object replacing Z is constructed with the following observations being taken into account (i) If (f, Ω) ∈ MΓ and x ∈ Ω is a solution to (3.1), then any y ∈ Γ(x) is also a solution to (3.1) Therefore, there is no any sense to speak about individual solutions to (3.1), and one should replace them with the corresponding orbits Moreover, if y1 , y2 ∈ Γ(x), then (Γy1 ) = (Γy2 ) = (Γx ) Hence, if an orbit Γ(x) belongs to the solution set to (3.1), then the conjugacy class (Γx ) completely characterizes symmetric properties of this orbit and the object replacing Z should strongly depend on Φ(Γ) (see Subsection 2.1) In addition, a simple transversality argument shows that generically, if (f, Ω) ∈ MΓ and f −1 (0) ⊃ Γ(x), then dim W (Γx ) = 0, i.e., W (Γx ) is finite (ii) In the nonsymmetric setting, if x1 and x2 are isolated solutions to equation (3.1), then their total contribution to deg(f, Ω) is equal to the sum of their independent contributions (see property (B2)), i.e., contributions of isolated solutions is compatible with the addition in Z If now (f, Ω) ∈ MΓ and Γ(x1 ), Γ(x2 ) ∈ Ω are in a solution set to (3.1), then the object replacing Z should contain an additive structure compatible with independent contributions of Γ(xi ) to the corresponding “Γ-equivariant degree.” (iii) Any Γ-symmetric generalization of property (B5) will appeal to Γ/Γx1 × Γ/Γx2 Observe that products of orbits of zeros Γ(x1 ) × Γ(x2 ) although any Γ-orbit admits only one orbit type, as a matter of fact, a product of two Γ-orbits usually contains several orbit types Therefore, the object replacing Z should admit a multiplication structure compatible with the “count” of orbits in products The Burnside ring we are going to define constitutes a formalization of the above stream of ideas Denote by A0 (Γ) = A(Γ) := Z[Φ0 (Γ)] the free abelian group generated by (H) ∈ Φ0 (Γ); i.e., an element a ∈ A(Γ) is a finite sum a = nH1 (H1 ) + · · · + nHm (Hm ) with nHi ∈ Z and (Hi ) ∈ Φ0 (Γ) Take (H), (K) ∈ Φ0 (Γ) and consider the diagonal Γ-action on Γ/H × Γ/K For any (L) ∈ Φ0 (Γ), the spaces Γ/H L and Γ/K L consist of finitely many W (L) orbits Therefore, the space (Γ/H × Γ/K)(L) /Γ is finite and one can define an operation of multiplication on A(Γ) by (H) · (K) = nL (H, K) (L), (3.3) (L)∈Φ0 (Γ) where nL (H, K) denotes the number of elements in the set (Γ/H×Γ/K)(L) /Γ; i.e., nL (H, K) := (Γ/H × Γ/K)(L) /Γ (3.4) In other words, the number nL (H, K) represents the number of orbits of type (L) contained in the space Γ/H × Γ/K In this way, A(Γ) becomes a ring with the unity (Γ) The ring A(Γ) is called the Burnside ring of Γ Author's personal copy 60 Z Balanov et al JFPTA Notice that dim W (H) > n implies Π(H) = {0} Indeed, for a ∈ Π(H)\{0}, one can choose a regular normal representative Fa satisfying (i)–(iii) Then, zero is a regular value of the W (H)-equivariant map FaH : B(Rm+n ⊕ V )H → −1 (Rm ⊕ W )H , and therefore (FaH ) (0) is an n-dimensional submanifold of m+n B(R ⊕ V )H composed of the W (H)-orbits of dimension smaller than or equal to n, i.e., dim W (H) ≤ n In this way, the values of degG (f, Ω) can be written as degG (f, Ω) = degG (f , Ω) = a(H1 ) + a(H2 ) + · · · + a(HN ) , a(Hi ) ∈ Π(Hi ) (7.4) Still, not all the components a(Hi ) of degG (f, Ω) can be easily computed Further simplifications of the equivariant degree are needed to create its more “computable” versions Definition 7.2 Let (f, Ω) be as in Assumption (f) Put Πp := Π(H) and dim W (H)=n Πs := Π(H) dim W (H) such that f (x) > ε for all x ∈ ∂Ω, and w1 , , wm ∈ E∞ with m F (Ω) ⊂ Nε := m Bε (wk ) = k=1 {w ∈ W : w − wk < ε}, k=1 where Bε (wk ) stands for the ball of radius ε centered at wk Define Pε : Nε → conv{w1 , , wm } by Pε (w) := m k=1 max{0, ε − w − wk } m max{0, ε − w − wk }wk , k=1 and put Fε := Pε ◦ F Next, apply the Haar integral to construct an equivariant finite-dimensional approximation Fε : Ω → W∗ ⊂ E (dim W∗ < ∞) of F , i.e., g Fε (g −1 x)dμ(g), Fε (x) := Fε (x) − F (x) < ε for all x ∈ Ω G Then, as in the nonequivariant case, degG (f, Ω) := degG (fε |Ω∗ , Ω∗ ) (fε := π − Fε and Ω∗ := Ω ∩ (Rn ⊕ W∗ )) is the Leray–Schauder equivariant degree (with n parameters) In a standard way, one can check that this degree is well defined, as well as formulate and derive its basic properties like existence, equivariant homotopy invariance, additivity, suspension, and the Hopf property 7.6 Construction of gradient G-degree: Finite-dimensional case Let G be a compact Lie group Take (f, Ω) ∈ MG ∇ (V, V ) (cf Section 5), and consider ε > such that < ε < ε(ϕ,∂Ω) := inf v∈∂Ω ∇ϕ(v) By applying Lemma B.3 (see Appendix B), choose a special Ω-Morse function ψ ∈ CG (V, R) such that sup ∇ϕ(v) − ∇ψ(v) < ε v∈Ω Since critical orbits of a special Ω-Morse function are isolated, the set of zeros of ∇ψ in Ω is composed of a finite number of regular critical orbits; i.e., there Author's personal copy Vol (2010) Equivariant degree theory 65 are v1 , , vk ∈ (∇ψ)−1 (0)∩Ω such that (∇ψ)−1 (0)∩Ω = G(v1 )∪· · ·∪G(vk ) Define the gradient G-degree of f in Ω according to the following formula: − − ∇G -deg (∇ϕ, Ω) = (−1)m1 (Gv1 ) + · · · + (−1)mk (Gvk ) ∈ U (G), m− i − (7.7) where := m (∇ ψ(vi )), i = 1, 2, , k (we not assume that (Gvj ) = (Gvi ) for i = j) The definition of the gradient equivariant degree was introduced by K G¸eba in [44], where it was proved that the above definition does not depend on the choice of the ε-approximating special Ω-Morse function ψ (in fact, G¸eba used in [44] a concept of generic functions, which is slightly stronger than the one used for special Ω-Morse functions) The fundamental properties (except the Hopf property) of the gradient degree were also proved in [44] 7.7 Gradient G-degree: Infinite-dimensional generalization Throughout this subsection, G is a compact Lie group, H a separable Hilbert G-representation with a fixed self-adjoint bounded G-equivariant Fredholm operator L : H → H of index zero Take a continuously differentiable Ginvariant functional η : H → R such that ∇η : H → H is a completely continuous operator, and define the functional Φ : H → R by Φ(u) := Lu, u − η(u), u ∈ H (7.8) Let Ω ⊂ H be an open G-invariant bounded subset such that (∇Φ, Ω) is an admissible pair In this subsection, following [48], we will outline the construction of a gradient G-degree ∇G -deg(∇Φ, Ω) ∈ U (G) (see also [94], where the case G = S was considered) To this end, we need the following standard definition Definition 7.11 A sequence of G-equivariant orthogonal projections Πn : H → H, n = 0, 1, 2, , is called an equivariant approximation scheme in H with respect to L (in short, an L-scheme) if (a) the subspaces Hn := Πn (H), n = 0, 1, 2, , are finite dimensional; (b) Hn ⊂ Hn+1 , n = 0, 1, 2, ; (c) limn→∞ Πn (u) = u for all u ∈ H; (d) Πn L = LΠn , n = 0, 1, 2, (in particular, L(Hn ) ⊂ Hn ); (e) H0 := ker L ⊂ H0 Remark 7.12 Notice that if ∞ H= Vk k=0 is the G-isotypical decomposition of H (with Vk being modelled on the irreducible representation Vk ) such that dim Vk < ∞ for all k = 0, 1, 2, , then the orthogonal projections on n Hn := Ker L ⊕ Vk , n = 0, 1, 2, , k=0 constitute an L-scheme Author's personal copy 66 Z Balanov et al JFPTA Take an L-scheme in H and put Z := ∇Φ−1 (0) ∩ Ω Since ∇Φ = L + ∇η is G-equivariant (cf (7.8)) and ∇η is completely continuous, it follows that Z is a compact G-invariant subset of Ω Put δ0 := dist (Z, ∂Ω) > and define for ε > the set Ωε := {u : dist (u, Z) < ε} It was shown in [48] that ∀ε< δ0 ∃n∈N ∀n>n u ∈ Ω δ0 \ Ωε =⇒ Lu = (Πn ◦ ∇η ◦ Πn )(u) 3 Therefore, for n > n, the gradient G-degree ∇G -deg (L − Πn ∇η, Ωε ∩ Hn ) is well defined provided ε is small enough In addition, since L is a Fredholm operator of index zero, the gradient G-degree ∇G -deg (L, B(Hn H0 )) is an invertible element in U (G) (here, Hn H0 := {u ∈ Hn : u, v = for all v ∈ H0 } and B(·) stands for the unit ball) We are now in a position to define the gradient G-degree of ∇Φ = L − ∇η on Ω by def ∇G -deg (L − ∇η, Ω) := (∇G -deg (L, B(Hn H0 )))−1 ∗ ∇G -deg (L − Πn ∇η, Ωε ∩ Hn ), (7.9) where n > n We refer the reader to [48] for the proofs that the above definition does not depend on choices of ε > 0, n > n or an L-scheme Remark 7.13 (i) One can easily show that, up to cosmetic modifications, the gradient G-degree defined by (7.9) satisfies properties similar to those listed in Theorem 5.2 (observe that admissible homotopies are of the form L(·) + K(t, ·), t ∈ [0, 1], where K is a completely continuous equivariant gradient map and L is fixed) (ii) In the case L = Id, one has ∇G -deg (L, B(Hn H0 )) = (G), and formula (7.9) coincides with the standard Leray–Schauder generalization of the finite-dimensional gradient G-degree (cf [96]) (iii) In general, when L = Id, the most problematic situation occurs when Φ is a strongly indefinite functional, i.e., dim H− = dim H+ = ∞ (here H + (resp., H − ) stands for the positive (resp., negative) subspace of L) This “capricious” setting was studied by many authors using different techniques (see, for example, [14, 16, 17, 18, 19, 23, 26, 27, 40, 45, 61, 71, 87, 86, 92] and the references therein) The crucial difficulty is related to the “conflict” between the existence of critical points of arbitrary large Morse index and the necessity to construct a suitable finite-dimensional reduction In the context relevant to our discussion, this means that the infinite negative spectrum of L gives rise to a sequence of basic degrees contributing the second factor in formula (7.9) that does not stabilize The first factor in formula (7.9) “neutralizes” these “contributions,” while the compactness of the perturbation of L preserves the stabilization of the degree in question Author's personal copy Vol (2010) Equivariant degree theory 67 Appendices A Regular normal maps and approximations As a matter of fact, for any (f , Ω) ∈ M, there exists (f, Ω) ∈ M such that (i) zero is a regular value of f (in particular, f −1 (0) ∩ Ω is finite), and (ii) f is as close as we need to f Since formula (3.2) allows (at least, in principal) to determine the Brouwer degree deg(f, Ω), one can use the same formula to compute deg(f , Ω) In particular, smooth maps transversal to zero perform as “nice representatives” of homotopy classes of maps related to M We arrive at the following question, what are the “nice representatives” of equivariant homotopy classes in the setting relevant to the (general) equivariant degree? Such a map f should have the following properties: (a) f separates zeros of different orbit types; (b) f −1 (0) consists of isolated orbits; (c) f behaves “regularly” around each orbit from f −1 (0) In the following definition, the normality is responsible for property (a), while the regular normality provides properties (a)–(c) together Definition A.1 Let V be an orthogonal G-representation and let (f, Ω) ∈ M(Rn ⊕ V, V ) with G acting trivially on Rn , Ω being invariant and f being equivariant (a) f is said to be normal in Ω if for each α = (H) ∈ Φ(G; Ω) and each x ∈ f −1 (0) ∩ ΩH , there exists a δx > such that for all w ∈ νx (Ωα ) with w < δx , f (x + w) = f (x) + w = w, where “νx (Ωα )” stands for the normal space to the manifold Ωα at x (b) f is said to be regular normal in Ω if (i) f is C -smooth; (ii) f is normal in Ω; (iii) for every (H) ∈ Φ(G, Ω), zero is a regular value of fH := f |ΩH : ΩH → W H Theorem A.2 Let (f , Ω) ∈ M(Rn ⊕ V, V ) with G acting trivially on Rn , Ginvariant Ω and G-equivariant f Then, for every η > 0, there exists (f, Ω) ∈ M such that f is regular normal in Ω and fulfills f (x) − f (x) < η for all x∈V Remark A.3 The notions of normality and regular normality can be reformulated in a canonical way for Ω-admissible G-equivariant homotopies h : [0, 1] × V → W , and again a regular normal approximation theorem holds Author's personal copy 68 Z Balanov et al JFPTA in this setting For applications it is of interest that if h joins regular normal maps h0 and h1 , then the regular normal approximation h of h can be chosen joining the original maps h0 and h1 B Special Morse functions Similarly to Appendix A, in Appendix B, we consider “nice representatives” of G-gradient Ω-admissible homotopies Lemma B.1 Let G be a compact Lie group and (∇ϕ, Ω) ∈ MG ∇ (V, V ) Then, for any v0 ∈ Ω, τv0 G(v0 ) V W ⊕ ∇2 ϕ(v0 ) : ⊕ v0 + W −→ H WH ⊕ ⊕ H ⊥ v0 + (W ) has the following form (W H )⊥ ⎡ ⎤ 0 ⎢ ∇2 ϕ(v0 ) = ⎢ ⎣ B(v0 ) 0 0 ⎥ ⎥, ⎦ (B.1) C(v0 ) where v0 + W is a normal space to the orbit G(v0 ) at v0 Definition B.2 Under the notations of Lemma B.1: (i) Given a v0 ∈ (∇ϕ)−1 (0) ∩ Ω, the critical orbit G(v0 ) is called regular if dim ker ∇2 ϕ(v0 ) = dim G(v0 ), i.e., the matrices B(v0 ), C(v0 ) (defined in (B.1)) are nonsingular (ii) The function ϕ is said to be a special Ω-Morse function, if for every v0 ∈ (∇ϕ)−1 (0) ∩ Ω, the critical orbit G(v0 ) is regular and the matrix C(v0 ) (defined in (B.1)) possesses only positive eigenvalues Definition B.2(ii) is independent of a choice of the point v0 Recall that for a symmetric matrix A, m− (A) denotes the total sum of dimensions of all the eigenspaces of A corresponding to negative eigenvalues Notice that if G(v0 ) ⊂ Ω is a critical orbit of a special Ω-Morse function ϕ, then m− (∇ϕ(v0 )) = m− (B(v0 )), where B is defined by (B.1) Lemma B.3 Let G be a compact Lie group and (∇ϕ, Ω) ∈ MG ∇ (V, V ) Then, for every > 0, there is a special Ω-Morse function ψ ∈ CG (V, R) such that sup ∇ϕ(x) − 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Wasserman, Equivariant differential topology Topology (1969), 127– 150 [102] J Wu, Theory and Applications of Partial Functional-Differential Equations Applied Mathematical Sciences 119, Springer, New York, 1996 Zalman Balanov Department of Mathematics and Computer Sciences Netanya Academic College Netanya 42365 Israel and Department of Mathematical Sciences University of Texas in Dallas EC 35, 800 West Campbell Road Richardson, TX 75080-3021 USA e-mail: balanov@mars.netanya.ac.il Author's personal copy 74 Z Balanov et al Wieslaw Krawcewicz Department of Mathematical Sciences University of Texas in Dallas EC 35, 800 West Campbell Road Richardson, TX 75080-3021 USA e-mail: wieslaw@utdallas.edu Slawomir Rybicki Faculty of Mathematics and Computer Science Nicolaus Copernicus University PL-87-100 Toru´ n ul Chopina 12/18 Poland e-mail: rybicki@mat.uni.torun.pl Heinrich Steinlein Mathematisches Institut Universită at Mă unchen Theresienstr 39 D-80333 Mă unchen Germany e-mail: steinl@mathematik.uni-muenchen.de JFPTA