A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

Abstract

Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of Hamiltonian systems, we establish a Bott-type iteration formula for dihedraly equivariant Hamiltonian systems. We apply the abstract theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Finally we provide a hyperbolicity criterion for reversible Lagrangian systems.

Appendices

A The cyclic and dihedral group algebras

Let G be a finite group with n elements, namely ${{\,\mathrm{ord}\,}}(G)=n$. We denote by ${\mathbf {C}}[G]$ the group algebra ofGover the complex field, namely the set of all formal complex linear combinations of element of G with coefficients in the complex field. It is well-know that ${\mathbf {C}}[G]$ is a ${\mathbf {C}}$-vector space of dimension ${{\,\mathrm{ord}\,}}(G)$ and it is a non-commutative algebra unless the group is commutative. In what follows we are interested in ${\mathbf {C}}[G]$-modules. An important example of ${\mathbf {C}}[G]$-module is ${\mathbf {C}}[G]$ itself viewed as a ${\mathbf {C}}[G]$-module and it is termed regular${\mathbf {C}}[G]$-module. As a direct consequence of the Maschke’s Theorem if V is a ${\mathbf {C}}[G]$-module then it is semisimple; thus there exist simple ${\mathbf {C}}[G]$-modules $U_1, \dots ,U_k$ such that

$$\begin{aligned} V= U_1 \oplus \dots \oplus U_k. \end{aligned}$$

In particular ${\mathbf {C}}[G]$ is semisimple. Moreover by the Schur’s Lemma, if V, W are simple ${\mathbf {C}}[G]$-modules either $\phi :V \rightarrow W$ is a ${\mathbf {C}}[G]$-isomorphism or $\phi$ is the trivial homomorphism. In the first case, $\phi$ is a scalar multiple of the identity map $I_V$. As direct consequence of Schur’s Lemma, if G is a finite abelian group, every simple ${\mathbf {C}}[G]$-module V is of length one. For, let $x,g \in G$ and $v \in V$; thus we have

$$\begin{aligned} x g\, v = g \, (xv) \qquad g, x \in G, \ \ v \in V. \end{aligned}$$

Therefore, $V \ni v \mapsto x\, v \in V$ is a ${\mathbf {C}}[G]$-homomorphism. Since V is irreducible (being simple), Schur’s Lemma implies that there exists $\lambda _x \in {\mathbf {C}}$ such that $xv= \lambda _x v$, for all $v \in V$. In particular, this implies that every subspace of V is a ${\mathbf {C}}[G]$-module and since V is simple, $\dim V=1$. (Actually, also the converse is true). As a direct consequence of the Wedderburn-Artin Theorem, we have that ${\mathbf {C}}[G]$ is isomorphic to the direct sum of matrix algebras over division rings (since we are working over an algebraic closed field, it turns out that each division rings is actually isomorphic to ${\mathbf {C}}$), there exists an isomorphism

$$\begin{aligned} \Phi : {\mathbf {C}}[G] \longrightarrow \mathrm {Mat}(n_1; {\mathbf {C}}) \oplus \dots \oplus \mathrm {Mat}( n_k; {\mathbf {C}}), \end{aligned}$$

such that ${{\,\mathrm{ord}\,}}(G)=\sum _{j=1}^k n_j^2$.

Example A.1

(The cyclic group) Let ${ C_{n}:=\langle r| r^n=e\rangle }$ be the cyclic group of order n. It can be realized as the group of rotations through angles $2k\pi /n$ around an axis. Denoting by ${\mathbf {C}}[z]$ the ring of complex polynomials, it readily follows that the group algebra ${\mathbf {C}}[ C_{n}]$ is isomorphic to the quotient of ${\mathbf {C}}[z]$ by the ideal generated by the n-th cyclotomic polynomial $z^n-1$; i.e.

$$\begin{aligned} {\mathbf {C}}[ C_{n}]\cong {\mathbf {C}}[z]/(z^n-1). \end{aligned}$$

Since we may factor the polynomial $z^n-1$ over the complex field as $z^n-1=\prod _{k=0}^{n-1} (z- \zeta _n^k)$, where $\zeta _n$ denotes a n-th root of unit, the group algebra splits as a product of n copies of ${\mathbf {C}}$ (exactly the Wedderburn-Artin decomposition), e.g.

$$\begin{aligned} {\mathbf {C}}[ C_{n}]= \prod _{k=0}^{n-1} {\mathbf {C}}[z]/(z-\zeta _n^k). \end{aligned}$$

Thus ${\mathbf {C}}[ C_{n}]$ decomposes as the direct sum of n one-dimensional (complex) subspaces, i.e.

$$\begin{aligned} {\mathbf {C}}[ C_{n}]= \bigoplus _{0 \leqslant k <n} L_k \end{aligned}$$

where $\dim _{\mathbf {C}}L_j=1$ for each $j=1, \dots ,n$. The generator t acts, in the corresponding factor, as multiplication by $\zeta _n^k$, meaning that $L_k$ is a $C_{n}$-module with respect to the following action $C_{n} \times L_k \rightarrow L_k: (t, w)\mapsto \zeta _n^k\, w$ and the group algebra ${\mathbf {C}}[ C_{n}]$ splits into the sum of n, $C_{n}$-modules.

Remark A.2

It is worth noticing that, by Example A.1 and by invoking the structure theorem of finite abelian groups, it readily follows the group algebra’s decomposition of any abelian group.

Example A.3

(The Dihedral group) Let $D_{n}$ be the dihedral group of degree n and order 2n. For $n \geqslant 3$, $D_{n}$ is the group of symmetries of a regular n-gon in the plane, namely, the group of all the plane symmetries that preserves a regular n-gon. It contains n rotations, which form a subgroup isomorphic to $C_{n}$ and n reflections. From an algebraic viewpoint it is a metabelian group having the cyclic normal subgroup $C_{n}$ of index 2 and the following presentation

$$\begin{aligned} D_{n}:=\langle r,s|r^n=s^2=e,\ srs=r^{-1}\rangle . \end{aligned}$$

In conclusion $D_{n}$ is a semi-direct product of $C_{n}$ and $C_{2}$; in symbols $D_{n}= C_{n} \rtimes C_{2}$, where $C_{n}:=\langle r| r^n=e\rangle$ and $C_{2}:=\langle s| s^2=e\rangle$. Each element of $D_{n}$ can be uniquely written, either in the form $r^k$, with $0 \leqslant k \leqslant n-1$ (if it belongs to $C_{n}$), or in the form $sr^k$, with $0 \leqslant k \leqslant n-1$ (if it doesn’t belong to $C_{n}$). Observe that the relation $srs=r^{-1}$ implies that $sr^ks=r^{-k}$ and $(sr^k)^2=e$. By the Wedderburn-Artin decomposition Theorem, the complex dihedral group algebra ${\mathbf {C}}[ D_{n}]$ splits as a product of complex matrices as follows

$$\begin{aligned} {\mathbf {C}}[ D_{n}]\cong \bigoplus _{d|n} M_d \end{aligned}$$

where $M_d \cong {\mathbf {C}}\oplus {\mathbf {C}}$ if $d=1,2$ and $M_d\cong \mathrm {Mat}(2; {\mathbf {C}})$ if $d >2$. Denoting by $L_k$ the one-dimensional complex subspace defined in Example A.1, we set ${{\widetilde{L}}}_k:=L_k \oplus L_{-k}$ and we identify $L_k$ with the horizontal subspace $L_k \times \{0\} \subset {{\widetilde{L}}}_k$ and $L_{-k}$ with the vertical, namely with $\{0\} \times L_{-k} \subset {{\widetilde{L}}}_k$. With the action $\phi : D_{n} \times {{\widetilde{L}}}_k \rightarrow {{\widetilde{L}}}_k$ defined by

$$\begin{aligned} \phi \big (r,(x,y)\big )= \big (\zeta _n\, x, \zeta _n^{-1}\, y\big ), \qquad \phi \big (s,(x,y)\big )= (y,x) \end{aligned}$$

${{\widetilde{L}}}_k$ is a $D_{n}$-module. As long as $k \not \equiv -k$ mod n, the module ${{\widetilde{L}}}_k$ is irreducible. On the contrary, if $n=2k$, then ${{\widetilde{L}}}_k= L_k \oplus L_k$ decomposes. Indeed, let $v_+\equiv (1,1)$ and $v_-\equiv (1,-1)$, then the subspaces $L_+:={\mathbf {C}}v_+$ and $L_-:={\mathbf {C}}v_-$ are invariant, so ${{\widetilde{L}}}_k$ decomposes as $L_+ \oplus L_-$. On $L_+$, the element s acts as the identity and r as the multiplication by $-1$, and on $L_{-}$ both act as multiplication by $-1$. Thus the group algebra decomposes in a direct sum of the $D_{n}$-modules as follows

$$\begin{aligned} {\mathbf {C}}[ D_{n}]= \bigoplus _{k=1}^{n_*}2\, {{\widetilde{L}}}_k\oplus {{\widetilde{L}}}_0 \oplus \delta \, {{\widetilde{L}}}_{n/2} \end{aligned}$$

where $\delta :=0$ if n is odd and 1 if n is even.

Let E be a separable complex Hilbert space and let $I_E$ be denote the identity operator in E. Given the unitary operator ${\mathcal {R}} \in {\mathbf {U}}(E)$, let ${\mathcal {C}}_n$ be the finite subgroup defined by ${\mathcal {C}}_n:=\langle {\mathcal {R}} \in {\mathbf {U}}(E)| {\mathcal {R}}^n = I_E\rangle \subset {\mathbf {U}}(E).$ (Thus ${\mathcal {C}}_n$ is isomorphic to the cyclic group $C_{n}$ and it is nothing but a unitary representation of the cyclic group $C_{n}$). By using the spectral mapping theorem it readily follows that the spectrum of ${\mathcal {R}}$ is respectively given by Let $E_k:=\ker ({\mathcal {R}}-\zeta _n^kI_E)$ be the eigenspace corresponding to the eigenvalue $\zeta _n^k$. By the spectral theory of normal operators, it is well-known that $E_k$ are mutually orthogonal and, clearly $C_{n}$ invariant, the action being given the multiplication by $\zeta _n^k$

$$\begin{aligned} C_{n} \times E_k \rightarrow E_k:(\zeta _n^{k}, v)\mapsto {\mathcal {R}}\,v=\zeta _n^{k}\,v. \end{aligned}$$

Thus, we get a orthogonal decomposition of the Hilbert space E into $C_{n}$-closed stable subspaces as follows

$$\begin{aligned} E=E_1 \oplus \dots \oplus E_n. \end{aligned}$$

(A.1)

Remark A.4

We observe that the decomposition given in Eq. (A.1) is the isotypic decomposition induced by the unitary representation of the cyclic group. In particular each subspace $E_k$ is given by the direct sum of (infinitely many) one-dimensional irreducible representations described in Example  A.1.

We denote by ${\mathcal {D}}_n$ the finite subgroup (actually a unitary representation of the dihedral group $D_{n}$) of the unitary group of E, presented by

$$\begin{aligned} {\mathcal {D}}_n :=\langle {\mathcal {R}}, {\mathcal {S}} \in {\mathbf {U}}(E)| {\mathcal {R}}^n= {\mathcal {S}}^2=({\mathcal {S}} {\mathcal {R}})^2= I_E\rangle \subset {\mathbf {U}}(E). \end{aligned}$$

For $k=0, \dots , n-1$, we define the closed linear subspaces $E_{k}:=\ker ({\mathcal {R}}-\zeta _n^kI_E)$ and $E_{-k}:=\ker ({\mathcal {R}}-\zeta _n^{-k}I_E)$ and for $k=1,\dots ,n_*$, we let

$$\begin{aligned} F_k:=E_k \oplus E_{-k}. \end{aligned}$$

(A.2)

We observe that $F_k$ defined in Eq. (A.2) is a $D_{n}$-module with the action given by

$$\begin{aligned} C_{n} \times F_k \rightarrow F_k:\big (\zeta _n^k,v\big )\longmapsto {\mathcal {R}}\, v=\begin{bmatrix} \zeta _n^{k}\, I_E&{}0\\ 0 &{} \zeta _n^{-k}\,I_E \end{bmatrix}\,v\quad \text { and } \\ C_{2} \times F_k \rightarrow F_k:\big (s,v\big )\longmapsto {\mathcal {S}}\, v=\begin{bmatrix} 0&{}I_E\\ I_E &{}0 \end{bmatrix}\,v. \end{aligned}$$

Thus, we get a decomposition of the Hilbert space E into mutually orthogonal $D_{n}$-stable modules, given by

$$\begin{aligned} E=\bigoplus _{k=0}^{{\bar{n}}} F_k. \end{aligned}$$

(A.3)

Remark A.5

We observe that the decomposition given in Eq. (A.3), is the isotypic decomposition of E with respect to the irreducible representations of $D_{n}$.

B Maslov index, spectral flow and index theorems

The goal of this section is to briefly recall the Definitions and the main Properties of the Maslov index (and its friends) and the Spectral flow. In order to make the presentation as smooth as possible, we split this Section into two different Subsections. In the first Subsection B.1 we start to recalling the differentiable structure of the Lagrangian Grassmannian of a real and complex symplectic space as well as some well-known facts about the (relative)$L_0$-Maslov index or more generally the Maslov index for paths of Lagrangian subspaces both on real and complex symplectic spaces. In order to make the Notations and Definitions as uniform as possible we start by recalling the differentiable structure of the Lagrangian Grassmannian which plays a crucial role in the description of the Maslov index through an intersection theory. Our basic references for the material contained in the first Subsection are [2, 7, 14, 17, 29, 35] and references therein. In Subsection B.2, starting on some useful functional analytic preliminaries and we continue to provide the construction as well as useful properties of the spectral flow for paths of closed selfadjoint Fredholm operators which are continuous in the gap topology, which we’ll need in the later Sections. Our basic references are [4,5,6, 18, 22, 28, 36].

1.1 B.1 The geometry of the Lagrangian Grassmannian and the Maslov index

Let $(V,\omega )$ be (finite) 2m-dimensional real symplectic vector space, ${\mathscr {L}}(V)$ be denote the vector space of all bounded and linear operators of V and let $J \in {\mathscr {L}}(V)$ be a complex structure compatible with the symplectic form $\omega$, meaning that $\omega (\cdot , J\cdot )$ is an inner product on V. Let us consider the Lagrangian Grassmannian of$(V,\omega )$, namely the set $\Lambda (V, \omega )$ and we recall that it is a real compact and connected analytic embedded $m(m+1)/2$-dimensional submanifold of the Grassmannian manifold of V. Moreover for any $L \in \Lambda (V,\omega )$ the tangent space $T_L\Lambda (V,\omega )$ is canonically isomorphic to the space $\mathrm {B}_{\text {sym}}(L)$ of all symmetric bilinear forms on L. Given $L_0\in \Lambda (V,\omega )$ and any non-negative integer , we define the sets $\Lambda ^j(L_0;V):=\big \{L\in \Lambda (V,\omega ):\dim (L\cap L_0)=k\big \}$ and we observe that $\Lambda (V,\omega ):=\bigcup _{j=0}^m \Lambda ^j(L_0;V)$. It is well-known that $\Lambda ^j(L_0;V)$ is a connected embedded analytic submanifold of $\Lambda (V,\omega )$ (being locally closed orbit of the Lie group $\mathrm {Sp}(V,\omega , L_0)$ of all symplectomorphism of $(V, \omega )$ which preserve $L_0$ (cfr. [38, Theorem 2.9.7]), and connected) having codimension equal to $j(j+1)/2$. In particular $\Lambda ^1(L_0;V)$ has codimension 1 and for $k \geqslant 2$ the codimension of $\Lambda ^j(L_0;V)$ in $\Lambda (V,\omega )$ is bigger or equal to 3. Its tangent space is canonically isomorphic to the space of all symmetric bilinear forms over L vanishing on $L \cap L_0$. A central object in our discussion is played by the universal Maslov (singular) cycle with vertex at$L_0$, being an algebraic (actually a determinantal) variety defined by

$$\begin{aligned} \Sigma (L_0;V) :=\bigcup _{j=1}^m \Lambda ^j(L_0;V). \end{aligned}$$

We observe that the Maslov cycle is the (topological) closure of lowest codimensional stratum $\overline{\Lambda ^1(L_0;V)}$. In particular, $\Lambda ^0(L_0;V)$, the set of all Lagrangian subspaces that are transversal to $L_0$, is an open and dense subset of $\Lambda (V,\omega )$. The (top stratum) codimensional 1-submanifold $\Lambda ^1(L_0;V)$ in $\Lambda (V,\omega )$ is co-oriented or otherwise stated it carries a transverse orientation. In fact given $\varepsilon >0$, for each $L \in \Lambda ^1(L_0;V)$, the smooth path of Lagrangian subspaces $\ell :(-\varepsilon , \varepsilon ) \rightarrow \Lambda (V,\omega )$ defined by $\ell (t):=\exp (tJ)$ crosses $\Lambda ^1(L_0;V)$ transversally. The desired transverse orientation is given by the direction along the path when the parameter runs between $(-\varepsilon , \varepsilon )$. In an equivalent way, the co-orientation or transverse orientation is meant in the following sense; we first observe that the mapping

$$\begin{aligned} T_L\Lambda (V,\omega ) \simeq \mathrm {B}_{\text {sym}}(L) \ni B \mapsto B\vert _{(L_0\cap L)\times (L_0\cap L)} \in \mathrm {B}_{\text {sym}}(L_0\cap L) \end{aligned}$$

passes to the quotient $T_L\Lambda (V,\omega )/T_L\Lambda ^j(L_0;V) \rightarrow \mathrm {B}_{\text {sym}}(L_0\cap L).$ The hypersurface $\Lambda ^1(L_0;V)$ carries a canonical transverse orientation which is defined by declaring that a vector $B\in T_L\Lambda (V,\omega ), B \notin T_L\Lambda ^1(L_0;V)$ is positively oriented if the non-zero symmetric bilinear form $B\vert _{(L\cap L_0)\times (L\cap L_0)}$ on the line $L\cap L_0$ is positive definite. Thus the Maslov cycle is two-sidedly embedded in $\Lambda (V,\omega )$. Based on these properties, Arnol’d in [2], defined an intersection index for closed loops in $\Lambda (V,\omega )$ (actually in $({\mathbf {R}}^{2m}, \omega )$, but the treatment in this more general situation presents no difficulties) via transversality arguments. This general position arguments can be generalised to Lagrangian paths (not only closed) with endpoints out of the Maslov cycle. Following authors in [7, 17] we introduce the following Definition.

Definition B.1

Let $L_0 \in \Lambda (V,\omega )$ and, for $a<b$, let $\ell \in {\mathscr {C}}^0\big ([a,b],\Lambda (V,\omega )\big )$. We define the (relative) Maslov index of$\ell$with respect to$L_0$ as the integer given by

$$\begin{aligned} \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, \ell ):=\left[ \exp (-\varepsilon J)\,\ell : \Sigma (L_0;V)\right] \end{aligned}$$

(B.1)

where $\varepsilon \in (0,1)$ is sufficiently small and where the right-hand side denotes the intersection number.

Remark B.2

A few Remarks on the Definition  B.1 are in order. By the basic geometric observation given in [7, Lemma 2.1], it readily follows that there exists $\varepsilon >0$ sufficiently small such that $\exp (-\varepsilon J)\,\ell (a), \exp (-\varepsilon J)\,\ell (b)$ doesn’t lie on $\Sigma (L_0;V)$. By [35, Step 2, Proof of Theorem 2.3], there exists a perturbed path ${{\widetilde{\ell }}}$ having only simple crossings (namely the path $\ell$ intersects the Maslov cycle transversally and in the top stratum. Since, simple crossings are isolated, on a compact interval are in a finite number. To each crossing instant $t_i \in (a,b)$ we associate the number $s(t_i)= 1$ (resp. $s(t_i)=-1$) according to the fact that, in a sufficiently small neighbourhood of $t_i$, ${{\widetilde{\ell }}}$ have the same (resp. opposite) direction of $\exp (t\,J) {{\widetilde{\ell }}}(t_i)$. Then the intersection number given in Formula (B.1) is equal to the summation of $s(t_i)$, where the sum runs over all crossing instants $s(t_i)$.

The Maslov index given in Definition B.1 have many important properties (cfr. [7, 35] for further dails). Below we list only a few of them that we’ll use in the sequel for computing the Maslov index and, for further details, we refer the interested reader to the aforementioned paper and references therein.

• Property I (Reparametrisation Invariance) Let $\psi :[a,b] \rightarrow [c,d]$ be a continuous function with $\psi (a)=c$ and $\psi (b)= d$. Then $\mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, \ell )=\mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, \ell \circ \psi ).$

• Property II (Homotopy invariance Relative to the Ends) Let

  $$\begin{aligned} {{\overline{\ell }}}: [0,1] \times [a,b] \rightarrow \Lambda (V,\omega ):(s,t)\mapsto {{\overline{\ell }}}(s,t) \end{aligned}$$

  be a continuous two-parameter family of Lagrangian subspaces such that $\dim \big (L_0\cap {{\overline{\ell }}}(s,a)\big )$ and $\dim \big (L_0\cap {\overline{\ell }}(s,b)\big )$ are independent on s. Then $\mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, {{\overline{\ell }}}_0) = \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, {{\overline{\ell }}}_1)$ where ${{\overline{\ell }}}_0(\cdot ):={{\overline{\ell }}}(0, \cdot )$ and ${{\overline{\ell }}}_1(\cdot ):={{\overline{\ell }}}(1, \cdot )$.

• Property III (Path Additivity) If $c \in (a,b)$, then

  $$\begin{aligned} \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, \ell )= \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, {{\overline{\ell }}}\vert _{[a,c]})+ \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, {{\overline{\ell }}}\vert _{[c,b]}). \end{aligned}$$

• Property IV (Symplectic Invariance) Let $\phi \in {\mathscr {C}}^0\big ([a,b], \mathrm {Sp}(V,\omega )\big )$ be a continuous path in the (closed) symplectic group $\mathrm {Sp}(V,\omega )$ of all symplectomorphisms of $(V,\omega )$. Then

  $$\begin{aligned} \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0, \ell )= \mu ^{\scriptscriptstyle {\mathrm {CLM}}}\big (\phi (t)\, L_0, \phi (t)\, \ell (t)\big ), \qquad t \in [0,1]. \end{aligned}$$

• Property V (Symplectic Additivity) For $i=1,2$ let $(V_i, \omega _i)$ be symplectic vector spaces, $L_i \in \Lambda (V_i,\omega _i)$ and let $\ell _i \in {\mathscr {C}}^0\big ([a,b], \Lambda (V_i,\omega _i\big )$. Then

  $$\begin{aligned} \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(\ell _1\oplus \ell _2, L_1\oplus L_2)= \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(\ell _1, L_1)+ \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(\ell _2, L_2). \end{aligned}$$

Although the Definition of the Maslov index given in Formula B.1 is apparently simple, the computation of this homotopy invariant is, in general, quite involved. One efficient technique for computing this invariant, was introduced (in the non-degenerate case) by the authors in [35] through the so-called crossing forms and, generalised (in the degenerate situation) by authors in [14, 15]. For $\varepsilon >0$ let $\ell ^* :(-\varepsilon , \varepsilon ) \rightarrow \Lambda (V,\omega )$ be a ${\mathscr {C}}^1$-path such that $\ell ^*(0)=L$. Let $L_1$ be a fixed Lagrangian complement of L and, for $v \in L$ and for sufficiently small t we define $w(t) \in L_1$ such that $v+w(t) \in \ell ^*(t)$. Then the form

$$\begin{aligned} Q[v]=\dfrac{d}{dt}\Big |_{t=0} \omega \big (v, w(t)\big ) \end{aligned}$$

(B.2)

is independent of the choice of $L_1$. A crossing instant$t_0$ for the continuous curve $\ell :[a,b] \rightarrow \Lambda (V,\omega )$ is an instant such that $\ell (t_0)\in \Sigma (L_0;V)$. If the curve is ${\mathscr {C}}^1$, at each crossing, we define the crossing form as the quadratic form on $\ell (t_0)\cap L_0$ given by

$$\begin{aligned} \Gamma (\ell , L_0, t_0)= Q(\ell (t_0, {{\dot{\ell }}}(t_0)\Big \vert _{\ell (t_0)\cap L_0} \end{aligned}$$

where Q was defined in Formula (B.2). A crossing $t_0$ is called regular if the crossing form is non-degenerate; moreover if the curve $\ell$ has only regular crossings we shall refer as a regular path. (Heuristically, $\ell$ has only regular crossings if and only if it is transverse to $\Sigma (L_0)$). Following authors in [29], if $\ell : [a,b] \rightarrow \Lambda (V,\omega )$ is a regular ${\mathscr {C}}^1$-path, then the crossing instants are in a finite number and the Maslov index is given by:

$$\begin{aligned} \mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L_0,\ell )=\mathrm {n_+}\left[ {\Gamma (\ell (a), L_0, a)}\right] + \sum _{\begin{array}{c} t_0 \in \ell ^{-1}\big (\Sigma (L_0;V)\big )\\ t_0 \in ]a,b[ \end{array}}{{\,\mathrm{sgn}\,}}\left[ \Gamma (\ell (t_0), L_0, t_0)\right] - \mathrm {n_-}{\left[ \Gamma (\ell , L_0, b)\right] }, \end{aligned}$$

where $\mathrm {n_+}, \mathrm {n_-}$ denotes respectively the number of positive (coindex), negative eigenvalues (index) in the Sylvester’s Inertia Theorem and where ${{\,\mathrm{sgn}\,}}:=\mathrm {n_+}-\mathrm {n_-}$ denotes the (signature). We observe that any ${\mathscr {C}}^1$-path is homotopic through a fixed endpoints homotopy to a path having only regular crossings.

In order to prove a dihedral equivariant Bott-type iteration formula by using the Maslov-type index, we need to review the Maslov index theory of the complex Lagrangian subspaces. For, let $({\mathbf {C}}^{2m}, \omega )$ be the complex symplectic vector space with the symplectic form $\omega (x,y)= \langle J x, y\rangle ,$ for all $x,y \in {\mathbf {C}}^{2m},$ where $\langle \cdot , \cdot \rangle$ denotes the standard Hermitian product in ${\mathbf {C}}^{2m}$, $J=\begin{bmatrix} 0 &{} -I_m\\ I_m &{} 0 \end{bmatrix}$ and $I_m$ is the identity matrix. If no confusion is possible we’ll omit the subindex m. We recall that a complex subspaceLis Lagrangian if $\omega |_L =0$ and the $\dim _{\mathbf {C}}L=m$. We let $L^\pm = \ker (iJ \mp {\mathbb {I}}_{2m})$ and we observe that L is a (complex) Lagrangian subspace L if and only if it can be seen as the graph of a unitary operator $U: L^+\rightarrow L^-$. Otherwise stated the complex Lagrangian Grassmannian is homeomorphic to the unitary group of ${\mathbf {C}}^{2m}$:

$$\begin{aligned} \Lambda ({\mathbf {C}}^{2m},\omega ) \cong {\mathbf {U}}(m). \end{aligned}$$

(B.3)

(Cfr. [3, 11, 34, 41] and references therein, for further details). We denote by ${\mathbb {F}}$ the homeomorphism defined in Formula (B.3) and we observe that

$$\begin{aligned} \dim \big (L_1 \cap L_2\big ) = \dim \ker \big ({\mathbb {F}}(L_2)^{-1} {\mathbb {F}}(L_1) -I_m\big ) . \end{aligned}$$

For any fixed $U \in {\mathbf {U}}(m)$, we let We refer to $\Sigma (U)$ as the singular cycle ofU. We observe that, for any $U_0 \in \Sigma (U)$, there exists $\varepsilon >0$ sufficiently small such that $e^{it} U_0$ is transversal to $\Sigma (U)$. Let now ${\mathcal {U}}:[a,b] \rightarrow {\mathbf {U}}(m)$ be a continuous path. For $\varepsilon >0$ small enough, $e^{-\varepsilon \, i} \, {\mathcal {U}}(a)$ and $e^{-\varepsilon \, i} \, {\mathcal {U}}(b)$ are out of the singular cycle of U and the intersection number of the perturbed path $e^{\varepsilon \,i}{\mathcal {U}}$ with the singular cycle $\Sigma (U)$, is well-defined. For this reason we are entitled to introduce the following Definition.

Definition B.3

Let $L \in \Lambda ({\mathbf {C}}^{2m}, \omega )$ be fixed and let $\ell :[a,b] \rightarrow \Lambda ({\mathbf {C}}^{2m}, \omega )$ be a continuous path. We define the (complex) Maslov index as $\mu ^{\scriptscriptstyle {\mathrm {CLM}}}(L, \ell ;[a,b]):=[e^{-\varepsilon \, i}{\mathbb {F}}(\ell ): \Sigma ({\mathbb {F}}(L)].$

Given $L \in \Lambda ({\mathbf {R}}^{2m}, \omega )$ be a real Lagrangian subspace, then $L^{\mathbf {C}}:=L \otimes {\mathbf {C}}\in \Lambda (C^{2m}, \omega )$. We define the manifold and we observe that $\Lambda _{\mathbf {R}}({\mathbf {C}}^{2m}, \omega )$ is isomorphic to $\Lambda ({\mathbf {R}}^{2m}, \omega ) \otimes {\mathbf {C}}$. It is worth noticing that, given $L \in \Lambda ({\mathbf {C}}^{2m}, \omega )$, we have ${\mathbb {F}}(e^{-\varepsilon J}L)= e^{-2 \varepsilon i} {\mathbb {F}}(L)$. Thus the (real) Maslov index given in Definition B.1 coincides with the (complex) Maslov index given in Definition B.3 when the path consists of real Lagrangian subspaces. So the Maslov index of a path of real Lagrangian subspaces is the same as the path of complex Lagrangian subspaces. We conclude by observing that the Definition of the crossing form as well as the computation of the Maslov index for a ${\mathscr {C}}^1$ regular path through crossing forms is the same as in the real case and it also fulfil Properties I-V given above.

1.2 B.2 On the spectral flow for paths of closed self-adjoint Fredholm operators

The aim of this Subsection is to briefly recall the Definition and the main properties of the spectral flow for a continuous path of closed self-adjoint Fredholm operator. It is well-known that this topological invariant was introduced by authors in [5] in order to develop an Index Theory on manifolds with boundary. Since then, it has been extensively applied and investigated extensively.

Let E be a separable complex Hilbert space and let $T: {\mathcal {D}}(T) \subset E \rightarrow E$ be a self-adjoint Fredholm operator. By the Spectral decomposition Theorem (cf., for instance, [20, Chapter III, Theorem 6.17]), there is an orthogonal decomposition $E= E_-(T)\oplus E_0(T) \oplus E_+(T),$ that reduces the operator T and has the property that

$$\begin{aligned} \sigma (T) \cap (-\infty ,0)=\sigma \big (T_{E_-(T)}\big ), \quad \sigma (T) \cap \{0\}=\sigma \big (T_{E_0(T)}\big ),\quad \sigma (T) \cap (0,+\infty )=\sigma \big (T_{E_+(T)}\big ). \end{aligned}$$

Definition B.4

Let $T \in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$. We term Tessentially positive if $\sigma _{ess}(T)\subset (0,+\infty )$, essentially negative if $\sigma _{ess}(T)\subset (-\infty ,0)$ and finally strongly indefinite respectively if $\sigma _{ess}(T) \cap (-\infty , 0)\not = \emptyset$ and $\sigma _{ess}(T) \cap ( 0,+\infty )\not = \emptyset$.

If $\dim E_-(T)<\infty$, we define its Morse index as the integer denoted by $\mu _{\scriptscriptstyle {\mathrm {Mor}}}\left[ T\right]$ and defined as $\mu _{\scriptscriptstyle {\mathrm {Mor}}}\left[ T\right] :=\dim E_-(T).$

Given $T \in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$, for $a,b \notin \sigma (T)$ we set ${\mathcal {P}}_{[a,b]}(T):=\mathrm {Re}\left( \dfrac{1}{2\pi \, i}\int _\gamma (\lambda -T)^{-1} d\, \lambda \right)$ where $\gamma$ is the circle of radius $\dfrac{b-a}{2}$ around the point $\dfrac{a+b}{2}$. We recall that if $[a,b]\subset \sigma (T)$ consists of isolated eigenvalues of finite type then

$$\begin{aligned} \mathrm {rge}\,{\mathcal {P}}_{[a,b]}(T)= E_{[a,b]}(T):=\bigoplus _{\lambda \in (a,b)}\ker (\lambda -T); \end{aligned}$$

(cf. [16, Section XV.2], for instance) and 0 either belongs in the resolvent set of T or it is an isolated eigenvalue of finite multiplicity. Let us now consider the graph distance topology which is the topology induced by the gap metric$d_G(T_1, T_2):=\left\| P_1-P_2 \right\|$ where $P_i$ is the projection onto the graph of $T_i$ in the product space $E \times E$. The next result allow us to define the spectral flow for gap continuous paths in ${{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$.

Proposition B.5

Let $T_0 \in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ be fixed.

• (i) There exists a positive real number $a \notin \sigma (T_0)$ and an open neighborhood ${\mathscr {N}} \subset {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ of $T_0$ in the gap topology such that $\pm a \notin \sigma (T)$ for all $T \in {\mathscr {N}}$ and the map

  $$\begin{aligned} {\mathscr {N}} \ni T \longmapsto {\mathcal {P}}_{[-a,a]}(T) \in {\mathscr {L}}(E) \end{aligned}$$

  is continuous and the projection ${\mathcal {P}}_{[-a,a]}(T)$ has constant finite rank for all $T \in {\mathscr {N}}$.

• (ii) If ${\mathscr {N}}$ is a neighborhood as in (i) and $-a \leqslant c \leqslant d \leqslant a$ are such that $c,d \notin \sigma (T)$ for all $T \in {\mathscr {N}}$, then $T \mapsto {\mathcal {P}}_{[c,d]}(T)$ is continuous on ${\mathscr {N}}$. Moreover the rank of ${\mathcal {P}}_{[c,d]}(T) \in {\mathscr {N}}$ is finite and constant.

Proof

For the proof of this result we refer the interested reader to [6, Proposition 2.10]. $\square$

Let ${\mathcal {A}}:[a,b] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ be a gap continuous path. As consequence of Proposition B.5, for every $t \in [a,b]$ there exists $a>0$ and an open connected neighborhood ${\mathscr {N}}_{t,a} \subset {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ of ${\mathcal {A}}(t)$ such that $\pm a \notin \sigma (T)$ for all $T \in {\mathscr {N}}_{t,a}$ and the map ${\mathscr {N}}_{t,a} \in T \longmapsto {\mathcal {P}}_{[-a,a]}(T) \in {\mathcal {B}}$ is continuous and hence ${{\,\mathrm{rank}\,}}\left( {\mathcal {P}}_{[-a,a]}(T)\right)$ does not depends on $T \in {\mathscr {N}}_{t,a}$. Let us consider the open covering of the interval [a, b] given by the pre-images of the neighborhoods ${\mathcal {N}}_{t,a}$ through ${\mathcal {A}}$ and, by choosing a sufficiently fine partition of the interval [a, b] having diameter less than the Lebesgue number of the covering, we can find $a=:t_0< t_1< \dots < t_n:=b$, operators $T_i \in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ and positive real numbers $a_i$, $i=1, \dots , n$ in such a way the restriction of the path ${\mathcal {A}}$ on the interval $[t_{i-1}, t_i]$ lies in the neighborhood ${\mathscr {N}}_{t_i, a_i}$ and hence the $\dim E_{[-a_i, a_i]({\mathcal {A}}_t)}$ is constant for $t \in [t_{i-1},t_i]$, $i=1, \dots ,n$.

Definition B.6

The spectral flow of${\mathcal {A}}$ (on the interval [a, b]) is defined by

$$\begin{aligned} {{\,\mathrm{sf}\,}}({\mathcal {A}}, [a,b]):=\sum _{i=1}^N \dim \,E_{[0,a_i]}({\mathcal {A}}_{t_i})- \dim \,E_{[0,a_i]}({\mathcal {A}}_{t_{i-1}}) \in {\mathbf {Z}}. \end{aligned}$$

(In shorthand Notation we denote ${{\,\mathrm{sf}\,}}({\mathcal {A}}, [a,b])$ simply by ${{\,\mathrm{sf}\,}}({\mathcal {A}})$ if no confusion is possible). The spectral flow as given in Definition B.6 is well-defined (in the sense that it is independent either on the partition or on the $a_i$) and only depends on the continuous path ${\mathcal {A}}$. (Cfr. [6, Proposition 2.13] and references therein). We list some useful properties of the spectral flow and we refer to [6] for further details.

• Property I (Path Additivity) If ${\mathcal {A}}_1: [a,b] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$, ${\mathcal {A}}_1,{\mathcal {A}}_2: [c,d] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ are two continuous path such that ${\mathcal {A}}_1(b)={\mathcal {A}}_2(c)$, then

  $$\begin{aligned} {{\,\mathrm{sf}\,}}({\mathcal {A}}_1 *{\mathcal {A}}_2) = {{\,\mathrm{sf}\,}}({\mathcal {A}}_1)+{{\,\mathrm{sf}\,}}({\mathcal {A}}_2). \end{aligned}$$

• Property II (Homotopy Relative to the Ends) If $h: [0,1]\times [a,b] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E):(s,t)\mapsto h(s,t)$ is a continuous map such that $h_a: [0,1] \ni s \mapsto \dim \ker h(s,a)\in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ and $h_b: [0,1]\ni t \mapsto \dim \ker h(s,b)\in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ are independent on s, then

  $$\begin{aligned} {{\,\mathrm{sf}\,}}(h_0,[a,b])={{\,\mathrm{sf}\,}}(h_1, [a,b]), \end{aligned}$$

  where $h_0(\cdot ):=h(0, \cdot )$ and $h_1(\cdot )=h(1,\cdot )$.

• Property III (Direct sum) If for $i=1,2$, $E_i$ are Hilbert spaces and if $h_i:[a,b] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E_i)$ are two gap-continuous paths of self-adjoint Fredholm operators, the

  $$\begin{aligned} {{\,\mathrm{sf}\,}}(h_1\oplus h_2,[a,b])= {{\,\mathrm{sf}\,}}(h_1,[a,b]) + {{\,\mathrm{sf}\,}}(h_2, [a,b]). \end{aligned}$$

As already observed, the spectral flow, in general, depends on the whole path and not just on the ends. However, if the path has a special form, it actually depends on the end-points. More precisely, let ${\mathcal {A}} ,{\mathcal {B}}\in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ and let $\widetilde{{\mathcal {A}}}:[a,b] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ be the path pointwise defined by $\widetilde{{\mathcal {A}}}(t):={\mathcal {A}}+ \widetilde{{\mathcal {B}}}(t)$ where $\widetilde{{\mathcal {B}}}$ is any continuous curve of ${\mathcal {A}}$-compact operators parametrised on [0, 1] such that $\widetilde{{\mathcal {B}}}(0):=0$ and $\widetilde{{\mathcal {B}}}(1):={\mathcal {B}}$. In this case, the spectral flow depends of the path ${{\widetilde{A}}}$, only on the endpoints (cfr. [28] and reference therein).

Remark B.7

It is worth noticing that, since every operator $\widetilde{{\mathcal {A}}}(t)$ is a compact perturbation of a a fixed one, the path $\widetilde{{\mathcal {A}}}$ is actually a continuous path into ${\mathscr {L}}({\mathcal {W}}; E)$, where ${\mathcal {W}}:={\mathcal {D}}({\mathcal {A}})$.

Definition B.8

([28, Definition 2.8]) Let ${\mathcal {A}} ,{\mathcal {B}}\in {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ and we assume that ${\mathcal {B}}$ is ${\mathcal {A}}$-compact (in the sense specified above). Then the relative Morse index of the pair${\mathcal {A}}$, ${\mathcal {A}}+{\mathcal {B}}$ is defined by

$$\begin{aligned} I({\mathcal {A}}, {\mathcal {A}}+{\mathcal {B}})=-{{\,\mathrm{sf}\,}}(\widetilde{{\mathcal {A}}};[a,b]) \end{aligned}$$

where $\widetilde{{\mathcal {A}}}:={\mathcal {A}}+ \widetilde{{\mathcal {B}}}(t)$ and where $\widetilde{{\mathcal {B}}}$ is any continuous curve parametrised on [0, 1] of ${\mathcal {A}}$-compact operators such that $\widetilde{{\mathcal {B}}}(0):=0$ and $\widetilde{{\mathcal {B}}}(1):={\mathcal {B}}$.

In the special case in which the Morse index of both operators ${\mathcal {A}}$ and ${\mathcal {A}}+{\mathcal {B}}$ are finite, then

$$\begin{aligned} I({\mathcal {A}}, {\mathcal {A}}+{\mathcal {B}})=\mu _{\scriptscriptstyle {\mathrm {Mor}}}\left[ {\mathcal {A}} +{\mathcal {B}}\right] -\mu _{\scriptscriptstyle {\mathrm {Mor}}}\left[ {\mathcal {A}}\right] . \end{aligned}$$

Let ${\mathcal {W}}, E$ be separable Hilbert spaces with a dense and continuous inclusion ${\mathcal {W}} \hookrightarrow E$ and let ${\mathcal {A}}:[0,1] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ having fixed domain ${\mathcal {W}}$. We assume that ${\mathcal {A}}$ is a continuously differentiable path ${\mathcal {A}}: [0,1] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}(E)$ and we denote by $\dot{{\mathcal {A}}}_{\lambda _0}$ the derivative of ${\mathcal {A}}_\lambda$ with respect to the parameter $\lambda \in [0,1]$ at $\lambda _0$.

Definition B.9

An instant $\lambda _0 \in [0,1]$ is called a crossing instant if $\ker \, {\mathcal {A}}_{\lambda _0} \ne 0$. The crossing form at $\lambda _0$ is the quadratic form defined by

$$\begin{aligned} \Gamma ({\mathcal {A}}, \lambda _0): \ker {\mathcal {A}}_{\lambda _0} \rightarrow {\mathbf {R}}, \quad \Gamma ({\mathcal {A}}, \lambda _0)[u] = \langle \dot{{\mathcal {A}}}_{\lambda _0}\, u, u\rangle _E. \end{aligned}$$

(B.4)

Moreover a crossing $\lambda _0$ is called regular, if $\Gamma ({\mathcal {A}}, \lambda _0)$ is non-degenerate.

We recall that there exists $\varepsilon >0$ such that ${\mathcal {A}} +\delta \, I_E$ has only regular crossings for almost every $\delta \in (-\varepsilon , \varepsilon )$. (Cfr., for instance [39, Theorem 2.6] and references therein). In the special case in which all crossings are regular, then the spectral flow can be easily computed through the crossing forms. More precisely the following result holds.

Proposition B.10

If ${\mathcal {A}}:[0,1] \rightarrow {{\mathcal {C}}}{{\mathcal {F}}}^{sa}({\mathcal {W}}, E)$ has only regular crossings then they are in a finite number and

$$\begin{aligned} {{\,\mathrm{sf}\,}}({\mathcal {A}}, [0,1]) = -\mathrm {n_-}{\left[ \Gamma ({\mathcal {A}},0)\right] }+ \sum _{t_0 \in (0,1)} {{\,\mathrm{sgn}\,}}\left[ \Gamma ({\mathcal {A}}, t_0)\right] + {\mathrm {n}}_{+}\left[ \Gamma ({\mathcal {A}},1)\right] \end{aligned}$$

where the sum runs over all the crossing instants.

Proof

The proof of this result follows by arguing as in [36]. This conclude the proof. $\square$