Common Singularities of Commuting Vector Fields

Abstract

We study the singularities of commuting vector fields of a real submanifold of a Kähler manifold Z.

1 Introduction

Let $(Z,\omega )$ be a connected Kähler manifold with a holomorphic action of a complex reductive group $U^\mathbb {C},$ where $U^\mathbb {C}$ is the complexification of a compact connected Lie group U with Lie algebra ${\mathfrak {u}}.$ We also assume $\omega$ is U-invariant and that there is a U-equivariant momentum map $\mu : Z \rightarrow {\mathfrak {u}}^*.$ By definition, for any $\xi \in {\mathfrak {u}}$ and $z\in Z,$ $d\mu ^\xi = i_{\xi _Z}\omega ,$ where $\mu ^\xi (z) := \mu (z)( \xi )$ and $\xi _Z$ denotes the fundamental vector field induced on Z by the action of U,  i.e.,

$$\begin{aligned} \xi _Z(z) := \frac{d}{dt}\bigg \vert _{t=0} \exp (t\xi )z \end{aligned}$$

(see, for example, Kirwan 1984 for more details on the momentum map). Since U is compact we may identify ${\mathfrak {u}} \cong {\mathfrak {u}}^*$ by means of a ${\textrm{Ad}}(U)$-invariant scalar product $\langle \cdot , \cdot \rangle$ on ${\mathfrak {u}}.$ Hence, we consider a momentum map as a ${\mathfrak {u}}$-valued map, i.e., $\mu : Z \rightarrow {\mathfrak {u}}.$ Recently, the momentum map has been generalized to the following settings (Heinzner and Schwarz 2007; Heinzner et al. 2008).

We say that a subgroup G of $U^\mathbb {C}$ is compatible if G is closed and the Cartan decomposition $U^\mathbb {C}=U\exp (\text {i} {\mathfrak {u}})$ induces a Cartan decomposition of G. This means that the map $K\times {\mathfrak {p}} \rightarrow G,$ $(k,\beta ) \mapsto k\text {exp}(\beta )$ is a diffeomorphism where $K := G\cap U$ and ${\mathfrak {p}} := {\mathfrak {g}}\cap \text {i}{\mathfrak {u}};$ ${\mathfrak {g}}$ is the Lie algebra of G. In particular K is a maximal compact subgroup of G with Lie algebra ${\mathfrak {k}}$ and ${\mathfrak {g}} = {\mathfrak {k}}\oplus {\mathfrak {p}}.$

Using $\langle \cdot , \cdot \rangle ,$ we define an ${\textrm{Ad}}(U)$-invariant scalar product on $\text {i}{\mathfrak {u}}$ requiring multiplication by $\text {i}$ to be an isometry between ${\mathfrak {u}}$ and $\text {i}{\mathfrak {u}}.$ The G-gradient map $\mu _{\mathfrak {p}}:Z \longrightarrow {\mathfrak {p}}$ associated with $\mu$ is the orthogonal projection of $\text {i}\mu$ onto ${\mathfrak {p}}.$ If $\beta \in {\mathfrak {p}}$ then

$$\begin{aligned} \mu _{\mathfrak {p}}^\beta (z) := \langle \mu _{\mathfrak {p}}(z), \beta \rangle = \langle \text {i}\mu (z), \beta \rangle = \langle \mu (z), -\text {i}\beta \rangle = \mu ^{-\text {i}\beta }(z), \end{aligned}$$

for any $\beta \in {\mathfrak {p}}$ and $z\in Z.$ In this paper, a G-invariant compact connected locally closed real submanifold X of Z is fixed and the restriction of $\mu _{\mathfrak {p}}$ to X is also denoted by $\mu _{\mathfrak {p}}.$ Then $\mu _{\mathfrak {p}}:X \longrightarrow {\mathfrak {p}}$ is a K-equivariant map such that $\text {grad}\mu _{\mathfrak {p}}^\beta = \beta _X,$ where the gradient is computed with respect to the induced Riemannian metric on X denoted by $(\cdot ,\cdot ).$ By the linearization Theorem (Heinzner et al. 2008; Sjamaar 1998), $\mu _{\mathfrak {p}}^\beta$ is a Morse–Bott function (Biliotti et al. 2013; Heinzner et al. 2008) and the limit

$$\begin{aligned} \varphi _\infty ^\beta (x):=\lim _{t\rightarrow +\infty } \exp (t\beta )x, \end{aligned}$$

exists and belongs to $X^\beta :=\{z\in X:\, \beta _X (z)=0\}$ for any $x\in X.$ The linearization theorem (Heinzner et al. 2008; Sjamaar 1998) also proves that any connected component of $X^\beta$ is an embedded submanifold, see for instance (Biliotti et al. 2013; Heinzner et al. 2008).

Let $C_1,\ldots , C_k$ be the connected components of $X^\beta .$ Let $W_i:=\{x\in X:\, \lim _{t\mapsto +\infty } \exp (t\beta )x \in C_i\}.$ Then $\mu _{\mathfrak {p}}^\beta (C_i)=c_i$ and applying again the linearization theorem (Heinzner et al. 2008; Sjamaar 1998), the submanifold $C_i$ is a connected component of $(\mu _{\mathfrak {p}}^\beta )^{-1} (c_i).$ One of the most important Theorem of Morse theory proves that $W_i$ is an embedded submanifold, which is called unstable manifold of the critical submanifold $C_i,$ and $\varphi _\infty ^\beta : W_i \longrightarrow C_i$ is smooth (Bott 1954).

Let T be a torus of U. This means that T is a connected compact Abelian subgroup of U (Adams 1969). By a Theorem of Koszul, (Duistermaat and Kolk 2000), the connected components of $Z^T:=\{x\in Z:\, T\cdot x=x\}$ are embedded Kähler submanifolds of Z. Let ${\mathfrak {t}}$ be the Lie algebra of T. It is well-known that the set

$$\begin{aligned} \left\{ \beta \in {\mathfrak {t}}:\, \overline{\exp (\mathbb {R}\beta )}=T \right\} , \end{aligned}$$

is dense in ${\mathfrak {t}},$ see for instance (Adams 1969). Hence,

$$\begin{aligned} Z^T=Z^{T^\mathbb {C}}=\left\{ p\in Z:\, \beta _Z(p)=0\right\} , \end{aligned}$$

(1)

for some $\beta \in {\mathfrak {t}}.$ This means $Z^T$ is the set of the singularities of the vector field $\beta _Z,$ i,e., the zero of the vector field $\beta _Z.$ Moreover, $Z^T$ is the image of the gradient flow $\varphi _\infty ^\beta$ defined by $\mu ^\beta .$

In this paper, we investigate the fixed point set of the action of an Abelian compatible subgroup of $U^\mathbb {C}$ acting on a real submanifold of Z.

Let ${\mathfrak {a}}\subset {\mathfrak {p}}$ be an Abelian subalgebra and $A=\exp ({\mathfrak {a}}).$ Notice that A is automatically closed in G and hence compatible, since $\{e\} \times {\mathfrak {a}}$ is closed in $K\times {\mathfrak {p}}.$ Then the A-gradient map on X is given by $\mu _{{\mathfrak {a}}}=\pi _{\mathfrak {a}}\circ \mu _{\mathfrak {p}},$ where $\pi _{\mathfrak {a}}:{\mathfrak {p}}\longrightarrow {\mathfrak {a}}$ denotes the orthogonal projection of ${\mathfrak {p}}$ onto ${\mathfrak {a}}.$ Since A is Abelian, then by Lemma 2.2 below for any $p\in X$ the stabilizer $A_p=\{a\in A:\, ap=p\}=\exp ({\mathfrak {a}}_p),$ where ${\mathfrak {a}}_p$ is the Lie algebra of $A_p.$ Therefore $X^A=\{p\in X:\, A\cdot p=p \}=\{ p\in X:\, \beta _X (p)=0,\, \forall \beta \in {\mathfrak {a}}\}.$ Hence, if $\alpha _1,\ldots ,\alpha _n$ is a basis of ${\mathfrak {a}}$ then $X^A$ is the set of the common singularities of the commuting vector fields $(\alpha _1)_X, \ldots , (\alpha _n)_X.$ Our first main result is the following

Theorem 1.1

The set $\left\{ \beta \in {\mathfrak {a}}:\, X^\beta =X^A \right\}$ is dense in ${\mathfrak {a}}.$

Hence $X^A$ is the set of the singularities of a vector field $\beta _X$ for some $\beta \in {\mathfrak {a}}$ and so the critical points of the Morse–Bott function $\mu _{\mathfrak {p}}^\beta .$

We point out that $X^A$ contains a lot of information of the geometry of both the A gradient map and the G gradient map. Indeed, for any $x\in X,$ $\mu _{\mathfrak {a}}(A\cdot x)$ is an open convex subset of $\mu _{\mathfrak {a}}(x)+{\mathfrak {a}}_x$ and $\overline{ \mu _{\mathfrak {a}}(A\cdot x) }={{\textrm{conv}}} (\mu _{\mathfrak {a}}(X^A \cap \overline{A\cdot x)}),$ see Atiyah (1982), Biliotti and Ghigi (2018) and Heinzner and Schützdeller (2010), where ${\textrm{conv}}(\cdot )$ denotes the convex hull of $(\cdot ).$ In particular $\mu _{\mathfrak {a}}(X^A)$ is a finite set and ${\textrm{conv}} (\mu _{\mathfrak {a}}(X))={\textrm{conv}} (\mu _{\mathfrak {a}}(X^A))$ and so a polytope. Moreover, if ${\mathfrak {a}}\subset {\mathfrak {p}}$ is a maximal Abelian subalgebra, then ${\textrm{conv}} (\mu _{\mathfrak {p}}(X))$ is given by $K{\textrm{conv}} ( \mu _{\mathfrak {a}}(X))$ (Biliotti et al. 2016).

The second main result proves the existence of $\beta \in {\mathfrak {a}}$ such that the limit map associated with the gradient flow of $\mu _{\mathfrak {p}}^\beta$ defines a map from X onto $X^A.$ Hence, the set $X^A$ is the image of the gradient flow of the Morse–Bott function $\mu _{\mathfrak {p}}^\beta$ for some $\beta \in {\mathfrak {a}}.$

Let $\alpha _1,\ldots ,\alpha _n\in {\mathfrak {a}}$ be a basis of ${\mathfrak {a}}.$ Then $\varphi _\infty ^{\alpha _n}\circ \cdots \circ \varphi _\infty ^{\alpha _1}$ defines a map from the manifold X onto $X^{\alpha _1} \cap \cdots \cap X^{\alpha _n}=X^A.$

Theorem 1.2

Let $\alpha _1,\ldots ,\alpha _n\in {\mathfrak {a}}$ be a basis of ${\mathfrak {a}}.$ There exists $\delta >0$ such that for any $0<\epsilon _2,\ldots ,\epsilon _n < \delta$ we have $\varphi _\infty ^{\alpha _1+\epsilon _2 \alpha _2 +\cdots + \epsilon _n \alpha _n}=\varphi _\infty ^{\alpha _n}\circ \cdots \circ \varphi _\infty ^{\alpha _1}.$

2 Proof of the Main Results

Suppose $X\subset Z$ is G-invariant compact connected real submanifold of Z with the gradient map $\mu _{\mathfrak {p}} : X\rightarrow {\mathfrak {p}}.$ If $x\in X$ then $G_x=\{g\in G:\, gx=x\}$ denotes the stabilizer of G at x. If $G_x$ acts on a manifold S,  then $G \times ^{G_x}S$ denotes the associated bundle with principal bundle $G \rightarrow G/G_x$ defined as the quotient of $G \times S$ by the $G_x$-action $h (g, s) = (gh^{-1}, h s).$ We recall the Slice Theorem, see Heinzner et al. (2008) for details.

Theorem 2.1

(Slice Theorem (Heinzner et al. 2008, Thm. 3.1; Sjamaar 1998) If $x \in X$ and $\mu _{\mathfrak {p}}(x) = 0,$ there are a $G_x$-invariant decomposition $T_x X = {\mathfrak {g}}\cdot x \oplus W,$ open $G_x$-invariant neighborhood S of $0 \in W,$ a G-stable open neighborhood $\Omega$ of $x\in X$ and a G-equivariant diffeomorphism $\Psi : G \times ^{G_x}S \rightarrow \Omega$ where $\Psi ([e, 0]) =x.$

Corollary 2.1.1

If $x \in X$ and $\mu _{\mathfrak {p}}(x) = \beta ,$ there are a $(G^\beta )_x$-invariant decomposition $T_x X = {\mathfrak {g}}^\beta \cdot x \, \oplus W,$ open $(G^\beta )_x$-invariant neighborhood S of $0\subset W,$ a $G^\beta$-stable open neighborhood $\Omega$ of $x\in X$ and a $G^\beta$-equivariant diffeomorphism $\Psi : G^\beta \times ^{{(G^\beta )}_x} S \rightarrow \Omega$ where $\Psi ([e, 0]) =x.$

This follows applying the previous theorem to the action of $G^\beta$ on X. Indeed, it is well known that $G^\beta =K^\beta \exp ({\mathfrak {p}}^\beta )$ is compatible (Biliotti et al. 2013, Lemma 2.7, p.584) and the orthogonal projection of $\text {i} \mu$ onto ${\mathfrak {p}}^\beta$ is the $G^\beta$-gradient map $\mu _{{\mathfrak {p}}^\beta }$ associated with $\mu$ (Heinzner et al. 2008). The group $G^\beta$ is also compatible with the Cartan decomposition of $(U^\mathbb {C})^{\beta }=(U^\mathbb {C})^{\text {i} \beta }=(U^{\text {i}\beta } )^\mathbb {C}$ and $\text {i} \beta$ is fixed by the $U^{\text {i}\beta }$-action on ${\mathfrak {u}}^{\text {i}\beta }.$ A momentum map of the $(U^\mathbb {C})^{\text {i} \beta }$-action on Z is given by $\widehat{\mu _{{\mathfrak {u}}^{\text {i}\beta }}(z)}=\pi _{{\mathfrak {u}}^{\text {i}\beta }} \circ \mu +\text {i}\beta ,$ where $\pi _{{\mathfrak {u}}^{\text {i}\beta }}$ is the orthogonal projection of ${\mathfrak {u}}$ onto ${\mathfrak {u}}^{\text {i}\beta },$ i.e., $U^{\text {i}\beta }$-shifted momentum map by an element of the center of ${\mathfrak {u}}^{\text {i}\beta }.$ Then, the associated $G^{\beta }$-gradient map with $\widehat{\mu _{{\mathfrak {u}}^{\text {i}\beta }}(z)}$ is given by $\widehat{\mu _{{\mathfrak {p}}^\beta }} := \mu _{{\mathfrak {p}}^\beta } - \beta$ and so $\widehat{\mu _{{\mathfrak {p}}^\beta }} (x)=0.$ Now, the result follows by Theorem 2.1. In particular, if G is commutative, then we have a Slice Theorem for G at every point of X,  see Heinzner et al. (2008, p.169) and Sjamaar (1998) for more details.

If $\beta \in {\mathfrak {p}},$ then $\beta _X$ is a vector field on X,  i.e. a section of the bundle TX. For $x\in X,$ the differential is a map $T_x X \rightarrow T_{\beta _X (x)}(TX).$ If $\beta _X (x) =0,$ there is a canonical splitting $T_{\beta _X (x)}(TX) = T_x X \oplus T_x X.$ Accordingly, the differential of $\beta _X,$ regarded as a section of TX,  splits into a horizontal and a vertical part. The horizontal part is the identity map. We denote the vertical part by $\mathrm d \beta _X (x).$ The linear map $\mathrm d \beta _X (x)\in {\text {End}}(T_x X)$ is indeed the so-called intrinsic differential of $\beta _X,$ regarded as a section in the tangent bundle TX,  at the vanishing point x. Let $\{\varphi _t=\exp (t\beta )\}$ be the flow of $\beta _X.$ There is a corresponding flow on TX. Since $\varphi _t(x)=x,$ the flow on TX preserves $T_x X$ and there it is given by $d\varphi _t(x) \in {\text {Gl}}(T_x X).$ Thus we get a linear $\mathbb {R}$-action on $T_x X$ given by $\mathbb {R}\times T_x X \longrightarrow T_x X,\, (t,v)\mapsto d\varphi _t(x) (v).$ The flow of the vector field $\beta _X$ defines an action of $\mathbb {R}$ on X,  i.e., $\mathbb {R}\times X \longrightarrow X,\, (t,x)\mapsto \exp (t\beta )x.$

Corollary 2.1.2

If $\beta \in {\mathfrak {p}}$ and $x \in X$ is a critical point of $\mu _{\mathfrak {p}}^\beta ,$ then there are open $\mathbb {R}$-invariant neighborhoods $S \subset T_x X$ and $\Omega \subset X$ and an $\mathbb {R}$-equivariant diffeomorphism $\Psi : S \rightarrow \Omega ,$ such that $0\in S, x\in \Omega ,$ $\Psi ( 0) =x.$ (Here $t\in \mathbb {R}$ acts as $d\varphi _t(x)$ on S and as $\varphi _t$ on $\Omega .)$

Proof

Since $\exp :{\mathfrak {p}}\longrightarrow G$ is a diffeomorphism onto the image, the subgroup $H:=\exp (\mathbb {R}\beta )$ is closed and so it is compatible. Hence, it is enough to apply the previous corollary to the H-action on X and the value at x of the corresponding gradient map.$\square$

Lemma 2.2

Let ${\mathfrak {a}}\subset {\mathfrak {p}}$ be an Abelian subalgebra and let $A=\exp ({\mathfrak {a}})$ which is closed and compatible. If $x\in X,$ then $A_x$ is compatible,  i.e.,  $A_x=\exp ({\mathfrak {a}}_x).$

Proof

If $a\in A_x,$ then $a=\exp (\beta )$ for a $\beta \in {\mathfrak {a}}.$ Let $f(t)=\langle \mu _{\mathfrak {a}}(\exp (t\beta )x), \beta \rangle .$ Then $f(1)= \langle \mu _{\mathfrak {a}}(\exp (\beta )x), \beta \rangle = \langle \mu _{\mathfrak {a}}(ax), \beta \rangle = \langle \mu _{\mathfrak {a}}(x), \beta \rangle = f(0)$ and $f'(t)=\parallel \beta _X (\exp (t\beta )x)\parallel ^2\ge 0 .$ This implies $\beta _X (x)=0$ and so $\beta \in {\mathfrak {a}}_x,$ proving $A_x = \exp ({\mathfrak {a}}_x).$ $\square$

Let $\alpha ,\beta \in {\mathfrak {p}}$ be such that $[\alpha ,\beta ]=0$ and let ${\mathfrak {a}}$ be the vector space in ${\mathfrak {p}}$ generated by $\alpha$ and $\beta .$ By the above Lemma, it follows that $X^A=X^\beta \cap X^\beta ,$ where $A=\exp ({\mathfrak {a}}),$ which is closed and compatible due to the fact that the exponential map is a diffeomorphism restricted on ${\mathfrak {p}}.$

Lemma 2.3

Let $\beta , \alpha \in {\mathfrak {p}}$ be such that $[\beta ,\alpha ] = 0.$ If X is compact,  then there exists $\delta > 0$ such that for any $\epsilon \in (0, \delta ),$ $X^{\beta + \epsilon \alpha } = X^\beta \cap X^\alpha .$

Proof

Let $\epsilon >0$ and let $A = \exp ({\mathfrak {a}}),$ where ${\mathfrak {a}}={\textrm{span}}(\alpha ,\beta ).$ Let $X^A$ denote the fixed point set of A,  i.e., $X^A=\{z\in X:\, A\cdot x=x\}.$ By Lemma 2.2, $X^A=X^\beta \cap X^\alpha .$ Corollary 2.1.2 applies for A and $H=\exp (\mathbb {R}(\alpha +\epsilon \beta )).$ Therefore $X^\beta \cap X^\alpha$ and $X^{\alpha +\epsilon \beta }$ are compact submanifolds satisfying $X^\beta \cap X^\alpha \subseteq X^{\alpha +\epsilon \beta }.$

Let C be a connected component of $X^\alpha \cap X^\beta .$ C is a compact connected submanifold of X and so it is arcwise connected. If $x\in C$ then $C\subseteq C',$ where $C'$ is the connected component of $X^{\alpha +\epsilon \beta }$ containing x. On the other hand, if L is a connected component of $X^{\alpha +\epsilon \beta }$ then L is A-stable and so there exists a A-gradient map (Heinzner et al. 2008). Since L is compact the norm square A-gradient map has a maximum. By Heinzner et al. (2008, Corollary 6.12) L has a fixed point of A. This implies that L contains a connected component of $X^{\alpha }\cap X^\beta .$ Summing up, we have proved that the number of the connected components of $X^\alpha \cap X^\beta$ is greater than or equal to the number of connected components of $X^{\alpha +\epsilon \beta }$ and any connected component of $X^{\alpha +\epsilon \beta }$ contains at least a connected component of $X^\alpha \cap X^\beta .$

Let $C_1,\ldots ,C_m$ be the connected components of $X^\alpha \cap X^\beta .$ Let $C_i'$ denote the connected component of $X^{\alpha +\epsilon \beta }$ containing $C_i,$ for $i=1,\ldots ,m.$ We point out that $C_k'$ would coincide with $C_j'$ for $k\ne i.$ We shall prove that there exists $\delta >0$ such that for any $\epsilon < \delta$ the connected components $C_1',\ldots ,C_m'$ are pairwise disjoints and $C_i=C_i'$ for $i=1,\ldots ,m.$

Let $x_i \in C_i.$ Since $x_i$ is fixed by A,  Corollary 2.1.2 implies there exists A-invariant open subsets $\Omega$ of $x_i\in X$ and S of $0\in T_{x_i} X$ and a A-equivariant diffeomorphism $\varphi : S \rightarrow \Omega$ such that $\varphi (0) = x_i,$ $d\varphi _0 = id_{T_{x_i} X}.$ Since the $\mathbb {R}$-action on S is linear and $\varphi$ is $\mathbb {R}$-equivariant, we may assume that $S=\Omega =\mathbb {R}^n$ by means of $\varphi ,$ $\alpha ,\beta$ are symmetric matrices of order n satisfying $[\alpha ,\beta ] = 0.$ Moreover, $T_{x_i} X^{\alpha +\epsilon \beta }={\textrm{Ker}}\, (\alpha +\epsilon \beta )$ and $T_{x_i} X^{\alpha }\cap T_{x_i} X^{\beta }={\textrm{Ker}}\, \alpha \cap {\textrm{Ker}}\, \beta .$

The matrices $\alpha$ and $\beta$ are simultaneously diagonalizable. Let $\{e_1,\ldots ,e_n\}$ be a basis of $\mathbb {R}^n$ such that $\alpha e_k =a_k e_k$ and $\beta e_k =b_k e_k$ for $k=1,\ldots , n.$ Let $J=\{1\le k \le n:\, a_k b_k \ne 0\}.$ Pick $\delta _{i} ={\textrm{min}}\{\frac{| a_k |}{| b_k |}:\, k\in J\}.$ Now, $(\alpha +\epsilon \beta )e_k=0$ if and only if $a_k+\epsilon b_k=0.$ If $a_k \ne 0,$ then $b_k \ne 0$ and vice-versa. If $\epsilon <\delta _i$ then $(\alpha +\epsilon \beta )e_k=0,$ if and only if $a_k=b_k=0.$ Therefore, ${\textrm{Ker}}\, (\alpha +\epsilon \beta )={\textrm{Ker}}\, \alpha \cap {\textrm{Ker}}\, \beta .$ This implies $T_{x_i} C_i = T_{x_i} C_i'.$ Although $\delta _i$ depends on $x_i,$ since $C_i\subseteq C_i'$ and both are compact submanifolds it follows that $C_i=C_i'.$ Pick $\delta ={\textrm{min}}(\delta _1,\ldots ,\delta _k).$ Then for any $\epsilon <\delta$ we have $C_i=C_i'$ for $i=1,\ldots ,m.$ In particular $C_1',\ldots ,C_m'$ are pairwise disjoints. Since the number of the connected components of $X^{\alpha +\epsilon \beta }$ is less than or equal to the number of connected components of $X^\alpha \cap X^\beta ,$ it follows that for any $\epsilon <\delta$ both $X^\alpha \cap X^\beta$ and $X^{\alpha +\epsilon \beta }$ have the same connected components and so $X^\alpha \cap X^\beta =X^{\alpha +\epsilon \beta }$ concluding the proof.$\square$

Theorem 2.4

Let ${\mathfrak {a}}\subset {\mathfrak {p}}$ be an Abelian subalgebra and let $A=\exp ({\mathfrak {a}}).$ Then the set

$$\left\{ \alpha \in {\mathfrak {a}}:\, X^A=X^\alpha \right\} $$

is dense.

Proof

Let $\alpha _1,\ldots ,\alpha _n$ be a basis of ${\mathfrak {a}}.$ Then

$$\begin{aligned} X^A=X^{\alpha _1} \cap \cdots \cap X^{\alpha _n}. \end{aligned}$$

By the above Lemma, there exists $\delta >0$ such that for any $\epsilon _2,\ldots ,\epsilon _n < \delta ,$ we have

$$\begin{aligned} X^A =X^{\alpha _1+ \epsilon _2 \alpha _2+\cdots +\epsilon _n \alpha _n} \end{aligned}$$

(2)

Let $\alpha \in {\mathfrak {a}}$ different form 0. It is well known that there exists $\alpha _2,\ldots \alpha _n\in {\mathfrak {a}}$ such that $\alpha ,\alpha _2,\ldots ,\alpha _n$ is a basis of ${\mathfrak {a}}.$ By (2), for any neighborhood U of $\alpha ,$ there exists $\beta \in U$ such that $X^A =X^\beta ,$ concluding the proof.$\square$

The following lemma is proved in Biliotti and Windare (2023), see also Bruasse and Teleman (2005, pag. 1036).

Lemma 2.5

Let $x\in X$ and $\beta , \alpha \in {\mathfrak {p}}$ be such that $[\beta , \alpha ]= 0.$ Set $y:= \lim _{t\rightarrow \infty }\exp (t\beta )x$ and $z:= \lim _{t\rightarrow \infty }\exp (t\alpha )y.$ Let $\delta$ be as in Lemma 2.3. Then for $0<\epsilon <\delta ,$

$$\begin{aligned} \lim _{t\rightarrow \infty }\exp (t(\beta + \epsilon \alpha ))x = z. \end{aligned}$$

As a consequence of the above lemma, we get the following result.

Theorem 2.6

Let $\alpha _1,\ldots ,\alpha _n$ be a basis of ${\mathfrak {a}}.$ Let $x\in X.$ Set $x_1:= \lim _{t\rightarrow \infty }\exp (t\alpha _1) x$ and $x_{i}= \lim _{t\rightarrow \infty }\exp (t\alpha _i) x_{i-1}$ for $i=2,\ldots ,n.$ Then there exists $\delta >0$ such that for $0<\epsilon _2,\ldots ,\epsilon _n<\delta ,$ we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\exp (t(\alpha _1 + \epsilon _2 \alpha _2 +\cdots + \epsilon _n \alpha _n))x = x_n, \end{aligned}$$

for any $x\in X.$ In particular,  $\varphi _\infty ^{\alpha _1+\epsilon _2 \alpha _2 +\cdots + \epsilon _n \alpha _n}=\varphi _\infty ^{\alpha _n}\circ \cdots \circ \varphi _\infty ^{\alpha _1}.$

Proof

By Theorem 2.4, there exists $\delta >0$ such that for any $0<\epsilon _2,\ldots ,\epsilon _n<\delta ,$ we have

$$\begin{aligned} X^A=X^{\alpha _1+\epsilon _2 \alpha _2+\cdots +\epsilon \alpha _n}. \end{aligned}$$

Let $A=\exp ({\mathfrak {a}}).$ Let $z\in X^A.$ By Corollary 2.1.2, there exists A-invariant open subsets $\Omega \subset X$ and $S\subset T_z X$ and a A-equivariant diffeomorphism $\varphi : S \rightarrow \Omega$ such that $0\in S,$ $z\in \Omega ,$ $\varphi (0) = z,$ $d\varphi _0 = id_{T_z X}.$ Let $x\in X.$ Set $x_1:= \lim _{t\rightarrow \infty }\exp (t\alpha _1) x$ and $x_{i}= \lim _{t\rightarrow \infty }\exp (t\alpha ) x_{i-1}$ for $i=2,\ldots ,n.$ If $x_n \in \Omega ,$ keeping in mind that $\Omega$ is A-invariant, it follows that $x_1,\ldots ,x_n\in \Omega .$ If one reads carefully the proof of Lemma 2.3, then $\delta >0$ works whenever that $y\in \Omega .$ The same argument applies in this case. Hence there exists $\delta$ such that for any $0<\epsilon _2,\ldots ,\epsilon _n <\delta ,$ we have

$$\begin{aligned} \lim _{t\mapsto } \exp (t(\alpha _1 + \epsilon _2\alpha _2+\cdots +\epsilon \alpha _n))x=x_n \end{aligned}$$

whenever $x_1,$ and so $x_1,\ldots ,x_n,$ belongs to $\Omega .$ By compactness of $X^A$ there exist open subsets $\Omega _1,\ldots ,\Omega _k$ satisfying the above property and such that

$$\begin{aligned} X^A \subseteq \Omega _1 \cup \cdots \cup \Omega _k. \end{aligned}$$

Let $\delta _1,\ldots .\delta _k$ as before. Pick $\delta ={\textrm{min}} (\delta _1,\ldots ,\delta _k).$ Let $x\in X.$ Set $x_1:= \lim _{t\rightarrow \infty }\exp (t\alpha _1) x$ and $x_{i}= \lim _{t\rightarrow \infty }\exp (t\alpha _i ) x_{i-1}$ for $i=2,\ldots ,n.$ Since $x_1\in \Omega _j$ for some $j=1,\ldots ,k,$ it follows that for any $0<\epsilon _2,\ldots ,\epsilon _n<\delta$ we have

$$\begin{aligned} \lim _{t\mapsto +\infty } \exp (t(\alpha _1 +\epsilon _2 \alpha _2 +\cdots +\epsilon _n \alpha _n))x=x_n. \end{aligned}$$

This holds for any $x\in X,$ concluding the proof.$\square$