Fixed points for nilpotent actions on the plane and the Cartwright–Littlewood theorem

Abstract

The goal of this paper is proving the existence and then localizing global fixed points for nilpotent groups generated by homeomorphisms of the plane satisfying a certain Lipschitz condition and having a bounded orbit. The Lipschitz condition is inspired in a classical result of Bonatti for commuting diffeomorphisms of the $2$-sphere and in particular it is satisfied by diffeomorphisms, not necessarily of class $C^{1}$, whose linear part at every point is uniformly close to the identity. In this same setting we prove a version of the Cartwright–Littlewood theorem, obtaining fixed points in any full continuum preserved by a nilpotent action.

Appendices

Appendix A: Revisiting Bonatti’s ideas

In order to show the existence of common fixed points for $C^1$-diffeomorphisms of the $2$-sphere, that are pairwise commuting and $C^{1}$-close to the identity, Bonatti studies their local properties [1]. In this section we adapt these results for homeomorphisms of the plane that are $\epsilon$-Lipschitz with respect to the identity. The proofs are essentially the same as in [1] and they are included in the paper for the sake of clarity.

Lemma 8.1

Let $\epsilon >0$, $n\in \mathbb {Z}^{+}$ and $f:\mathbb {R}^2\rightarrow \mathbb {R}^2$ with $\hbox {Lip}(f-Id)\le \epsilon /n$. Consider $p,q\in \mathbb {R}^2$ such that $\Vert q-p\Vert \le n \Vert f(p)-p\Vert$. Then we have

$$\begin{aligned} \Vert (f-Id)(q)-(f-Id)(p)\Vert \le \epsilon \Vert f(p)-p\Vert . \end{aligned}$$

(8.1)

In particular if $0<\epsilon < 1$ and $f(p)\ne p$ then

• $(i)$ $f$ has no fixed points in the closed ball $B\big [p, n\Vert f(p)-p\Vert \big ];$

• $(ii)$ The angle $\hbox {Ang}(v_{1},v_{2})$ enclosed by the vectors

  $$\begin{aligned} v_{1}=\frac{f(z_{1})-z_{1}}{\Vert f(z_{1})-z_{1}\Vert } \quad \text {and} \quad v_{2}=\frac{f(z_{2})-z_{2}}{\Vert f(z_{2})-z_{2}\Vert } \end{aligned}$$

  (8.2)

  is well-defined if $z_{1},z_{2}\in B\big [p, n\Vert f(p)-p\Vert \big ]$ and $0\le \hbox {Ang}(v_{1},v_{2})\le 2 \arcsin (\epsilon )$.

Proof

Let $p,q\in \mathbb {R}^2$ with $\Vert q-p\Vert \le n \Vert f(p)-p\Vert$. The Lipschitz property implies

$$\begin{aligned} \Vert (f-Id)(q)-(f-Id)(p)\Vert \le \frac{\epsilon }{n} \Vert q-p\Vert \le \frac{\epsilon }{n} n \Vert f(p)-p\Vert = \epsilon \Vert f(p)-p\Vert \end{aligned}$$

proving the inequality (8.1).

Suppose $0<\epsilon < 1$ and $f(p)\ne p$. If $f(q)=q$ for some $q\in B\big [p, n\Vert f(p)-p\Vert \big ]$ then

$$\begin{aligned} \Vert f(p)-p\Vert =\Vert (f-Id)(q)-(f-Id)(p)\Vert \le \epsilon \Vert f(p)-p\Vert \end{aligned}$$

contradicting the condition $0<\epsilon < 1$. This completes the proof of item $(i)$.

Let us show item $(ii)$. The angle enclosed by the vectors $v_{1}$ and $v_{2}$ is well-defined by item $(i)$. We have

$$\begin{aligned} \hbox {Ang}(v_{1},v_{2})\le \hbox {Ang}\left( v_{1},\frac{f(p)-p}{\Vert f(p)-p\Vert }\right) + \hbox {Ang}\left( v_{2},\frac{f(p)-p}{\Vert f(p)-p\Vert }\right) . \end{aligned}$$

Moreover Eq. (8.1) implies

$$\begin{aligned} 0\le \sin \bigg \{\hbox {Ang}\left( v_{i},\frac{f(p)-p}{\Vert f(p)-p\Vert }\right) \bigg \}\le \frac{\Vert f(z_{i})-z_{i}-\left( f(p)-p\right) \Vert }{\Vert f(p)-p\Vert } \le \epsilon . \end{aligned}$$

Thus we obtain $0\le \hbox {Ang}(v_{1},v_{2})\le 2 \arcsin (\epsilon )$, completing the proof of item $(ii)$.$\square$

Corollary 8.2

Let $f:\mathbb {R}^2\rightarrow \mathbb {R}^2$ with $\hbox {Lip}(f-Id)\le \frac{1}{8}$ and $p\in \mathbb {R}^2-\hbox {Fix}(f)$. Then

• $(i)$ $f$ is a homeomorphism$;$

• $(ii)$ $\Vert (f-Id)(q)-(f-Id)(p)\Vert \le \frac{1}{2} \Vert f(p)-p\Vert$ for all $p,q \in B[p,4\Vert f(p)-p\Vert ];$

• $(iii)$ $f$ has no fixed points in the closed ball $B[p,4\Vert f(p)-p\Vert ];$

• $(iv)$ $0\le \hbox {Ang}(v_{1},v_{2})\le \pi /3$ where $v_{1},v_{2}$ are defined in (8.2) and $z_{1},z_{2}$ belong to the closed ball $B[p,4\Vert f(p)-p\Vert ]$.

Proof

The proof of items $(ii), (iii)$ e $(iv)$ is obtained by considering $\epsilon =1/2$ and $n=4$ in Lemma 8.1. Item $(i)$ is an immediate consequence of Lemma 3.3.$\square$

The next lemma is used in the Proof of Lemma 6.1.

Lemma 8.3

Let $f\in \hbox {Homeo}(\mathbb {R}^2)$ with $\hbox {Lip}(f-Id)\le \frac{1}{8}$. Suppose that $p\in \mathbb {R}^2-\hbox {Fix}(f)$ is an $\omega$-recurrent point for $f$. Then $\Gamma ^{f}_{\!\!p}$ is not a simple curve.

Proof

Suppose that $\Gamma ^{f}_{\!\!p}$ is simple. Let $\sigma$ be a line segment, intersecting $[p,f(p)]$ perpendicularly in their common midpoint. We also suppose that the length of $\sigma$ is less or equal than $\Vert f(p)-p\Vert$. Since $p$ is $\omega$-recurrent the curve $\Gamma ^{f}_{\!\!p}$ intersects $\sigma$ infinitely many times.

Let $j \in \mathbb {Z}^{+}$ such that $\big [f^{j}(p),f^{j+1}(p)\big ] \cap \sigma \ne \emptyset$. We have

$$\begin{aligned} \Vert f^{j}(p)-p\Vert&\le \Vert f^{j+1}(p)-f^{j}(p)\Vert + 2 \Vert f(p)-p\Vert \\ {}&\le 3 \max \big \{ \Vert f(p)-p\Vert , \Vert f^{j+1}(p)-f^{j}(p)\Vert \big \}. \end{aligned}$$

Hence either $f^{j}(p)\in B[p,4\Vert f(p)-p\Vert ]$ or $p\in B[f^{j}(p), 4\Vert f^{j+1}(p)-f^{j}(p)\Vert ]$. Anyway, the angle defined by the segments $[p,f(p)]$ and $[f^{j}(p),f^{j+1}(p)]$ is less or equal than $\pi / 3$ by item $(iv)$ in Corollary 8.2. Therefore these segments intersect $\sigma$ with the same orientation.

Let $i\in \mathbb {Z}^{+}$ be the first positive integer such that $\big [f^{i}(p),f^{i+1}(p)\big ) \cap \sigma \ne \emptyset$. Consider the simple closed curve $\beta$ obtained by juxtaposing the segments

$$\begin{aligned}{}[a,f(p)], \ [f(p),f^{2}(p)], \ \ldots , [f^{i-1}(p),f^{i}(p)], \ [f^{i}(p),b], \ [b,a] \end{aligned}$$

where $\{a\} = [p,f(p)] \cap \sigma$ and $\{b\} = \big [f^{i}(p),f^{i+1}(p)\big ) \cap \sigma$. Since the segments $[p,f(p)]$ and $\big [f^{i}(p),f^{i+1}(p)\big )$ intersect $\sigma$ with the same orientation then $\beta$ separates the points $p$ and $f^{i+1}(p)$.

Let $\mathcal {D}$ be the closure of the connected component of $\mathbb {R}^2- \beta$ containing $f^{i+1}(p)$. Since $p$ and $f^{i+1}(p)$ belong to the $\omega$-limit of ${\mathcal O}_{p}(f)$ there exists $q =f^{j}(p)$ for some $j > i+1$ such that $q \in \hbox {Int}(\mathcal {D})$ and $f(q)\notin \hbox {Int}(\mathcal {D})$. The intersection of $[q,f(q)]$ with $\beta$ is contained in the segment $[a,b]$. Since $[p,f(p)]$ and $[q,f(q)]$ intersect $\sigma$ with the same orientation we deduce $q$ does not belong to $\hbox {Int}(\mathcal {D})$, obtaining a contradiction.$\square$

Let $p\in \mathbb {R}^2-\text {Fix}(f)$ be an $\omega$-recurrent point for $f$. There exists a simple closed curve $\gamma \subset \Gamma ^{f}_{\!\!p}$ by Lemma 8.3. The vertices of $\gamma$ are intersections of line segments of the form $[f^{n}(p),f^{n+1}(p)]$ and $[f^{m}(p),f^{m+1}(p)]$. The angle described by two such intersecting segments is less or equal than $\pi /3$ by Corollary 8.2. Let $\mathcal {D}$ be the disc bounded by $\gamma$. Consider the vector field defined by $X(x)=f(x)-x$ for $x\in \mathbb {R}^2$. Since the singularities of $X$ are the fixed points of $f$ the vector field $X$ has no singular points in $\gamma$ by Corollary 8.2. Moreover, the angle described by $X(x)$ and any segment of $\gamma$ through $x$ is less or equal than $\pi /3$. Therefore the index of the singularities of $X$ in $\mathcal {D}$ is equal to $1$. We obtain the following lemma:

Lemma 8.4

Let $f \in \hbox {Homeo}(\mathbb {R}^2)$ with $\hbox {Lip}(f-Id)\le \frac{1}{8}$. Consider a $\omega$-recurrent point $p\in \mathbb {R}^2-\hbox {Fix}(f)$ for $f$. Then there exists a simple closed curve $\gamma \subset \Gamma ^{f}_{\!\!p}$, contained in $\mathbb {R}^2- \hbox {Fix}(f)$, such that the compact set of fixed points of $f$ in the interior of the disc bounded by $\gamma$ has index $1$ for $f$.

Analogously the existence of an increasing sequence of positive integers $(n_{k})_{k\ge 1}$ such that $f^{n_{k}}(p) \rightarrow p$ for some $p\in \mathbb {R}^2-\hbox {Fix}(f)$ implies that a simple closed curve $\gamma _{k}\subset \Gamma ^{f}_{p,n_{k}}$ behaves similarly as the curve $\gamma$ in Lemma 8.4 if $k >>0$. The reason is that the segment $[f^{n_{k}}(p),f(p)]$ that is contained in $\Gamma ^{f}_{\!\!p,n_{k}}$ but it is not necessarily contained in $\Gamma ^{f}_{\!\!p}$ has analogous properties as the segment $[p,f(p)]$ when $k >>0$. Then we have the following lemma.

Lemma 8.5

Let $f \in \hbox {Homeo}(\mathbb {R}^2)$ with $\hbox {Lip}(f-Id)\le \frac{1}{8}$. Suppose that there exist $p\in \mathbb {R}^2-\hbox {Fix}(f)$ and an increasing sequence of positive integer numbers $(n_{k})_{k\ge 1}$ such that $f^{n_{k}}(p) \rightarrow p$. Suppose further that $\gamma _{k}\subset \Gamma ^{f}_{\!\!p,n_{k}}$ is a simple closed curve. Then $\gamma _{k}$ has no fixed points of $f$ and the compact set of fixed points of $f$ in the interior of the disc bounded by $\gamma _{k}$ has index $1$ for $f$ when $k>>0$.

We can provide versions of the results of this paper for the sphere $\mathbb {S}^{2} \subset \mathbb {R}^{3}$. We can adapt the proofs of Theorem 1.1 of [7] and Bonatti’s Main Theorem of [1] to the $\epsilon$-Lipschitz with respect to the identity setting. We remind the reader that if $f\in \hbox {Homeo}(\mathbb {S}^{2})$ is $\epsilon$-Lipschitz with respect to the identity then $f$ is $C^{0}$-close to the identity map when $\epsilon >0$ is small enough.

Appendix B: Some examples

There exist several results in the literature showing existence and localization of fixed points of abelian groups. In this section we show that the scope of our results is wider. More precisely, we provide examples of non-abelian nilpotent groups satisfying the conditions in the main theorem, i.e. $\sigma$-step nilpotent subgroups of homeomorphisms that are $\epsilon$-Lipschitz with respect to the identity, and admitting a global fixed point.

Examples arising from dimension $1$

We build examples of nilpotent subgroups of homeomorphisms of the plane that are inspired by examples of groups of homeomorphisms of the real line or the circle.

Plante and Thurston proved that the $C^{2}$-regularity imposes strong restrictions on the nilpotent groups of diffeomorphisms of the real line [20].

Theorem

(Plante–Thurston) Every nilpotent subgroup of $\hbox {Diff}^{2}([0,1])$ or $\hbox {Diff}^{2}([0,1))$ is abelian.

Farb–Franks generalize the previous result for $\hbox {Diff}^{2}(S^{1})$ in [8]. Moreover they prove the following version for diffeomorphisms of the line.

Theorem

(Farb–Franks) The groups of diffeomorphisms of the line satisfy:

• There exist $\sigma$-step nilpotent subgroups of $\hbox {Diff}^{\infty }(\mathbb {R})$ for any $\sigma \ge 0 ;$

• Any nilpotent subgroup of $\hbox {Diff}^{2}(\mathbb {R})$ is metabelian, i.e. $G_{(1)}$ is abelian$;$

• If $G$ is a nilpotent subgroup of $\hbox {Diff}^{2}(\mathbb {R})$ and if any element of $G$ has fixed points, then $G$ is abelian.

In contrast, the case $C^{1}$ is radically different.

Theorem

(Farb–Franks) Let $M=\mathbb {R}, S^{1}$ or $[0,1]$. Every finitely generated torsion-free nilpotent group is isomorphic to a subgroup of $\hbox {Diff}^{1}(M)$.

Given a group $G$ we denote by $\hbox {Tor}(G)$ the subset of $G$ of elements of finite order. In general $\hbox {Tor}(G)$ is not a group but it is a normal subgroup of $G$ if $G$ is a nilpotent group (cf. [17, Theorem 16.2.7]). We say that $G$ is torsion-free if $\hbox {Tor}(G)=\{Id\}$.

We denote by $\mathcal {N}_{n}$ the subgroup of $\hbox {GL}(n,{\mathbb Z})$ of lower triangular matrices such that all coefficients in the main diagonal are equal to $1$. The group $\mathcal {N}_{n}$ is $(n-1)$-step nilpotent for any $n \ge 2$. Given $1\le i< n$ we denote by $\eta _{i}$ the matrix in $\mathcal {N}_{n}$ such that the coefficient in the location $(i+1,i)$ is equal to $1$ and all other coefficients outside of the main diagonal vanish. The family $\{\eta _{i}\}_{1\le i<n}$ is a generator set of $\mathcal {N}_{n}$. A theorem of Malcev (cf. [21]) guarantees that a finitely generated torsion-free nilpotent group is isomorphic to a subgroup of $\mathcal {N}_{n}$ for some $n \ge 1$.

Theorem 2.13 in [8] and its proof imply the following result.

Theorem

(Farb–Franks) Let $\epsilon >0$. There exists an injective homomorphism of groups

$$\begin{aligned} \psi :\mathcal {N}_{n} \longrightarrow \hbox {Diff}^{1}([0,1]) \end{aligned}$$

such that:

• any element of $\psi (\mathcal {N}_{n})$ has derivative equal to $1$ in both $0$ and $1;$

• $|\psi (\eta _{i})'(x)-1|<\epsilon$ in $[0,1]$ for all $1\le i<n$.

Let us define a monomorphism $\Psi :\mathcal {N}_{n} \longrightarrow \hbox {Diff}^{1}(\mathbb {R}^2)$. Given $\eta \in \mathcal {N}_{n}$ let $\Psi (\eta )$ be the map defined by

$$\begin{aligned} \Psi (\eta )(t(x,y)):={\left\{ \begin{array}{ll} [\psi (\eta )(t)](x,y) &{} \mathrm if \quad 0 \le t \le 1 \\ \ (x,y) &{} \mathrm if \quad t>1 \end{array}\right. } \end{aligned}$$

where $(x,y) \in {\mathbb S}^{1}$ and $t \ge 0$. Moreover, we can suppose that $\psi (\eta _{i})'$ is arbitrarily close to the constant function $1$ for any $1 \le i \le n-1$. The nilpotent group $\Psi (\mathcal {N}_{n})$ is generated by the family ${\{ \Psi (\eta _{i}) \}}_{1 \le i <n}$, whose elements are $C^{1}$-close to the identity. Therefore any finitely generated torsion-free nilpotent group can be realized as a subgroup of $\hbox {Diff}^{1}(\mathbb {R}^2)$ whose generators are arbitrarily close to the identity in the $C^{1}$-topology.

Navas proves the following result in a recent paper (cf. [19, Théorème A]).

Theorem

(Navas) Let $G$ be a finitely generated subgroup of sub-exponential growth of orientation-preserving homeomorphisms of $[0,1]$ or $S^{1}$. Then, given any $\epsilon >0$, there exist subgroups that are topologically conjugated to $G$ and such that the generators and their inverses are $e^{\epsilon }$-Lipschitz homeomorphisms.

The above theorem implies that in the one dimensional setting the Lipschitz property can be assumed for the study of the topological dynamics of finitely generated nilpotent groups of homeomorphisms. Finitely generated nilpotent groups have polynomial growth. A simple calculation shows that the generators in the above theorem are $(e^{\epsilon }-1)$-Lipschitz with respect to the identity.

Real analytic examples

Our goal is providing examples of groups of real analytic diffeomorphisms of the plane that have a global fixed point, any nilpotency class and generators arbitrarily close to the identity map in the $C^{1}$-topology. Indeed we find nilpotent Lie algebras of real analytic vector fields in ${\mathbb S}^{2}$ whose set of common singular points is the set $\{0,\infty \}$ for any nilpotency class. Our examples are finitely generated subgroups of the image by the exponential map of such Lie algebras. Notice that all the above examples are essentially one dimensional and share the constraints of the one dimensional theory. For instance the previous section does not provide an example of a non-abelian nilpotent subgroup $G$ of $\hbox {Diff}^{2}({\mathbb R}^{2})$ with a global fixed point since such groups do not exist in dimension $1$. We show that such two dimensional examples exist for any nilpotency class and real analytic regularity.

We obtain other interesting results. It is well known that Lie algebras of real analytic vector fields defined in surfaces are metabelian. It was not known if there are examples of Lie algebras of real analytic vector fields in ${\mathbb S}^{2}$ of any nilpotency class. We prove that this is the case and along the way we give examples of torsion-free nilpotent groups of real analytic diffeomorphisms in the sphere for any nilpotency class.

We denote by $\hbox {Diff}^{\omega }(S)$ the group of real analytic diffeomorphisms defined in a real analytic manifold $S$. Let $\hbox {Diff}_{+}^{\omega }(S)$ be the subgroup of $\hbox {Diff}^{\omega }(S)$ of orientation-preserving diffeomorphisms.

Let ${\mathfrak g}$ be a Lie algebra. We denote ${\mathfrak g}_{(0)}=\mathfrak g$ and let ${\mathfrak g}_{(j+1)}$ be the Lie algebra generated by $\big \{ [f,g]; \ f \in {\mathfrak g} \ \hbox {and} \ g \in {\mathfrak g}_{(j)} \big \}$ for $j \ge 0$. We say that ${\mathfrak g}$ is $\sigma$-step nilpotent if $\sigma$ is the first element in ${\mathbb Z}_{\ge 0}$ such that ${\mathfrak g}_{(\sigma )}=\{0\}$. In that case we say that $\sigma$ is the nilpotency class of ${\mathfrak g}$. We say that ${\mathfrak g}$ is nilpotent if it is $\sigma$-step nilpotent for some $\sigma \in {\mathbb Z}_{\ge 0}$.

A Lie algebra ${\mathfrak g}$ of real analytic vector fields in a surface is always metabelian, i.e. ${\mathfrak g}_{(1)}$ is abelian. Moreover the nilpotent subgroups of $\hbox {Diff}^{\omega }({\mathbb S}^{2})$ are metabelian by a theorem of Ghys [13]. In spite of this, the nilpotency class is not bounded. The dihedral group

$$\begin{aligned} D_{2^{\sigma }} = \langle f,g ; \ f^{2^{\sigma }} = 1, g^{2}=1 \ \ \text {and} \ \ g \circ f \circ g^{-1} = f^{-1} \rangle \end{aligned}$$

is a $\sigma$-step nilpotent group acting by Mobius transformations on the sphere. The subgroup $\langle f \rangle$ is an index $2$ abelian normal subgroup of $D_{2^{\sigma }}$. An analogous property always holds true for nilpotent groups of $C^{1}$-diffeomorphisms that preserve both area and orientation.

Theorem 9.1

(Franks–Handel [10]) Let $G$ be a nilpotent subgroup of $\hbox {Diff}_{+}^{1}({\mathbb S}^{2})$. Let $\mu$ be a $\phi$-invariant Borel probability measure for any $\phi \in G$. Suppose that the support of $\mu$ is the whole sphere. Then either $G$ is abelian or it contains an index $2$ normal abelian subgroup.

In particular the above theorem implies that a torsion-free nilpotent subgroup of $\hbox {Diff}^{1}_{+}({\mathbb S}^{2})$ preserving a measure of total support is always abelian. We consider groups of real analytic diffeomorphisms instead of the area preserving hypothesis. We are replacing a rigidity condition with another one and it is natural to ask if analyticity restricts the examples of nilpotent groups as much as preservation of area. Indeed Ghys suggests in [13] that the quotient $G/\hbox {Tor}(G)$ is likely to be of nilpotency class less or equal than $2$ for any nilpotent subgroup $G$ of $\hbox {Diff}^{\omega }({\mathbb S}^{2})$. We will show that this is not the case.

Theorem 9.2

Given $\sigma \in {\mathbb Z}^{+}$ there exists a $\sigma$-step nilpotent torsion-free subgroup of $\hbox {Diff}_{+}^{\omega }({\mathbb S}^{2})$.

We also obtain:

Theorem 9.3

Given $\sigma \in {\mathbb Z}^{+}$ there exist $\phi _{1},\ldots ,\phi _{\sigma +1} \in \hbox {Diff}_{+}^{\omega }({\mathbb R}^{2})$ sharing a common fixed point and such that $\langle \phi _{1}, \ldots , \phi _{\sigma +1} \rangle$ is $\sigma$-step nilpotent. Moreover, we can choose the generators $\phi _{j}$ arbitrarily and uniformly close to the identity in the $C^{1}$-topology.

A method to obtain nilpotent Lie algebras. Let us explain our method for the real plane. We denote by ${\mathbb R}[x,y]$ the ring of real polynomials in two variables. Let $\alpha _{1},\beta _{1}$ be quotients of convenient elements of ${\mathbb R}[x,y]$ such that:

• (1) $d \alpha _{1} \wedge d \beta _{1} = D dx \wedge dy$ where $1/D \in {\mathbb R}[x,y]$;

• (2) $\frac{1}{D} d \alpha _{1}$ and $\frac{1}{D}d \beta _{1}$ are $1$-forms with no poles in ${\mathbb R}^{2}$.

Let us consider vector fields $X_{1}$ and $Y_{1}$ defined in ${\mathbb R}^{2}$ such that

$$\begin{aligned} \left\{ \begin{array}{ccc} X_{1}(\alpha _{1}) &{} \equiv &{} 0 \\ Y_{1}(\alpha _{1}) &{} \equiv &{} 1 \end{array} \right. \ \ \text {and} \ \ \left\{ \begin{array}{ccl} X_{1}(\beta _{1}) &{} \equiv &{} 1 \\ Y_{1}(\beta _{1}) &{} \equiv &{} 0. \end{array} \right. \end{aligned}$$

A straightforward calculation implies

$$\begin{aligned} X_{1} = \frac{-\frac{\partial \alpha _{1}}{\partial y}}{D} \frac{\partial }{\partial x} + \frac{\frac{\partial \alpha _{1}}{\partial x}}{D} \frac{\partial }{\partial y} \quad \text {and} \quad Y_{1} = \frac{\frac{\partial \beta _{1}}{\partial y}}{D} \frac{\partial }{\partial x} + \frac{-\frac{\partial \beta _{1}}{\partial x}}{D} \frac{\partial }{\partial y} . \end{aligned}$$

Condition (2) implies that $\frac{1}{D} d \alpha _{1}$ is polynomial. Since $\frac{1}{D} d \alpha _{1}$ is equal to $\frac{1}{D} \frac{\partial \alpha _{1}}{\partial x} dx + \frac{1}{D} \frac{\partial \alpha _{1}}{\partial y} dy$ then $X_{1}$ is a polynomial vector field. Analogously $Y_{1}$ is a polynomial vector field. Moreover $\alpha _{1}$ and $\beta _{1}$ can be considered as variables outside of a proper real algebraic set. We obtain $[X_{1},Y_{1}] \equiv 0$ since $[X_{1},Y_{1}](\alpha _{1}) \equiv [X_{1},Y_{1}](\beta _{1}) \equiv 0$ by definition.

Therefore the real vector space $\langle X_{1}, \alpha _{1} X_{1}, \ldots , \alpha _{1}^{l-1} X_{1}, Y_{1}\rangle _{\mathbb R}$ is a $l$-step nilpotent Lie algebra of polynomial vector fields of ${\mathbb R}^{2}$ if $\alpha _{1}^{l-1} X$ has no poles.

Let us introduce the choice of $\alpha _{1}$ and $\beta _{1}$ that is going to provide our examples. Consider

$$\begin{aligned} \left( \alpha _{1}, \beta _{1}, d \alpha _{1} \wedge d \beta _{1} \right) = \left( \frac{1}{(x^{2}+y^{2})^{k}} , \frac{y}{x (x^{2}+y^{2})^{p}} , -2 k \frac{dx \wedge dy}{x^{2} (x^{2}+y^{2})^{k+p}} \right) \ \ \text {with} \ \ k,p\ge 1. \end{aligned}$$

We obtain

$$\begin{aligned} X_{1} = x^{2} (x^{2}+y^{2})^{p-1}\left\{ -y \frac{\partial }{\partial x} + x \frac{\partial }{\partial y} \right\} \end{aligned}$$

and

$$\begin{aligned} Y_{1}= - \frac{1}{2k} (x^{2}+y^{2})^{k-1} \left\{ x\left( x^{2} + (1-2p) y^{2}\right) \frac{\partial }{\partial x} + y\left( (1+2p) x^{2} + y^{2}\right) \frac{\partial }{\partial y} \right\} . \end{aligned}$$

We also have that $(d \alpha _{1})/D$ and $(d \beta _{1})/D$ has no poles. Moreover if $p-1 \ge k(l-1)$ then the vector field $\alpha _{1}^{l-1} X_{1}$ has no poles.

In summary, $\langle X_{1}, \alpha _{1} X_{1}, \ldots , \alpha _{1}^{l-1} X_{1}, Y_{1}\rangle _{\mathbb R}$ is a $l$-step nilpotent Lie algebra of polynomial vector fields if $p-1 \ge k(l-1)$.

Examples of nilpotent groups on the sphere. Let us try to generalize the example to the sphere. Unfortunately the vector fields $X_{1}$ and $Y_{1}$ do not extend to real analytic vector fields of ${\mathbb S}^{2}$.

Let us study the dynamics of $X_{1}$ and $Y_{1}$. Since $X_{1}(\alpha _{1}) \equiv 0$ the function $x^{2}+y^{2}$ is a first integral of $X_{1}$. The trajectories of $X_{1}$ are contained in circles and $\hbox {Sing} (X_{1}) = \{x=0\}$. On the other hand the trajectories of $Y_{1}$ are transversal to the level curves of $x^{2}+y^{2}$ since $Y_{1} (\alpha _{1}) \equiv 1$. This condition also implies that $\hbox {exp}(t Y_{1})$ sends $\{\alpha _{1} =s \}$ to $\{\alpha _{1} =s+t \}$ for $s,t \in {\mathbb R}$.

We can interpret a point $(x,y)$ of ${\mathbb R}^{2}$ as an element $z$ of ${\mathbb C}$ via the identification $z=x+iy$. We will use the complex notation since it simplifies the presentation. Consider an annulus

$$\begin{aligned} A = \{z \in {\mathbb C} ; R^{-1} < |z| <R \} = \{ (x,y) \in {\mathbb R}^{2} ; \ R^{-1} < \sqrt{x^{2}+y^{2}} < R \} \end{aligned}$$

for some $R >1$. We define $X_{2} = - (1/z)_{*} X_{1}$ and $Y_{2} = - (1/z)_{*} Y_{1}$; both vector fields are defined in ${\mathbb S}^{2} - \{ |z| \le R^{-1} \}$. We claim that there exists a real analytic diffeomorphism $\phi : A \rightarrow A$ such that $\phi _{*} X_{1} = X_{2}$ and $\phi _{*} Y_{1} = Y_{2}$. More precisely $\phi$ conjugates the pair of functions $(\alpha _{1}, \beta _{1})$ with

$$\begin{aligned} \left( - \alpha _{1} \circ \frac{1}{z} + \kappa , - \beta _{1} \circ \frac{1}{z} \right) = \left( - (x^{2} + y^{2})^{k} +\kappa , \frac{y (x^{2} + y^{2})^{p}}{x} \right) \end{aligned}$$

where $\kappa =R^{2 k} + R^{-2 k}$. The sets $\alpha _{1}(A)$ and $(- \alpha _{1} \circ (1/z) + \kappa )(A)$ coincide by definition of $\kappa$. The map $(\alpha _{1},\beta _{1})$ is not injective since $(\alpha _{1},\beta _{1})(x,y) = (\alpha _{1},\beta _{1})(-x,-y)$. Anyway a homeomorphism $\phi : \overline{A} \rightarrow \overline{A}$ is uniquely determined by the conditions

$$\begin{aligned} \left( - \alpha _{1} \circ \left( \frac{1}{z} \right) + \kappa \right) \circ \phi \equiv \alpha _{1}; \quad - \beta _{1} \circ \left( \frac{1}{z} \right) \circ \phi \equiv \beta _{1} \quad \text {and} \quad \phi \{x>0\} \subset \{x>0\} \end{aligned}$$

Since $\alpha _{1}$, $\beta _{1}$ are coordinates in the neighborhood of any point in $x \ne 0$ then the map

$$\begin{aligned} \phi : A - \{x=0\} \rightarrow A - \{x=0\} \end{aligned}$$

is a real analytic diffeomorphism. Analogously $\alpha _{1}$ and $1/\beta _{1}$ are real analytic coordinates in the neighborhood of the points in the line $x=0$. We deduce that $\phi :A \rightarrow A$ is a real analytic diffeomorphism.

Let $S$ be the real analytic surface $S$ obtained by pasting the charts $U_{1} = \{|z| < R\}$ and $U_{2} = \{|z| > R^{-1}\}$ by the diffeomorphism $\phi$. The surface $S$ is homeomorphic to a sphere, so it is real analytically diffeomorphic to a sphere. Since $\phi _{*} X_{1} =X_{2}$ there exists a unique real analytic vector field $X$ in ${\mathbb S}^{2}$ such that $X_{|U_{1}} \equiv X_{1}$ and $X_{|U_{2}} \equiv X_{2}$. Analogously we can define $Y, \alpha$ and $\beta$ such that $Y_{|U_{1}} \equiv Y_{1}$, $Y_{|U_{2}} \equiv Y_{2}$, $\alpha _{|U_{1}} \equiv \alpha _{1}$, $\alpha _{|U_{2}} \equiv - \alpha _{1} \circ (1/z) + \kappa$, $\beta _{|U_{1}} \equiv \beta _{1}$ and $\beta _{|U_{2}} \equiv - \beta _{1} \circ (1/z)$. We obtain that

$$\begin{aligned} {\mathcal G} := \langle X, \alpha X, \ldots , \alpha ^{l-1} X, Y \rangle _{\mathbb R} \end{aligned}$$

is a $l$-step nilpotent Lie algebra of real analytic vector fields of the sphere. In particular ${\mathcal G}$ is non-abelian if $l >1$.

The vector field $Y$ has two singularities corresponding to the origin of both local charts. We can suppose $\hbox {Sing}(Y) = \{0,\infty \}$. The common singular set of all vector fields of the form $P(\alpha ) X$, where $P$ is a polynomial of degree less than $l$, is a circle $C$. The equation of $C$ is $x=0$ in both charts. Hence all vector fields in ${\mathcal G}$ are singular at both $0$ and $\infty$. We obtain that ${\mathcal G}$ is a Lie algebra of real analytic vector fields defined in both ${\mathbb R}^{2}$ and ${\mathbb S}^{2}$.

The Baker-Campbell-Hausdorff formula. Given a Lie algebra ${\mathcal G}$ as defined above the set $G=\hbox {exp}({\mathcal G})$ of time $1$ flows of elements of ${\mathcal G}$ is a subset of $\hbox {Diff}_{+}^{\omega }({\mathbb S}^{2})$. In this section we show that $G$ is a nilpotent group of the same nilpotency class as ${\mathcal G}$. This is an application of Baker–Campbell–Hausdorff formula. The material in this section is well-known, it is included for the sake of completeness.

Let $G$ be a Lie group with Lie algebra ${\mathfrak g}$. The exponential $\hbox {exp}({\mathfrak g})$ does not coincide in general with the connected component of the identity but anyway it contains a neighborhood of the identity element. Hence given elements $\hbox {exp}(Z)$ and $\hbox {exp}(W)$ in $G$ closed to the identity element we have that the element $\hbox {exp}(Z) \hbox {exp}(W)$ belongs to $\hbox {exp}({\mathfrak g})$ and $\log (\hbox {exp}(Z) \hbox {exp}(W))$ is given by the following formula due to Dynkin:

$$\begin{aligned} \sum _{n>0}\frac{(-1)^{n-1}}{n} {\mathop {\mathop {\sum }\limits _{r_i + s_i > 0}}\limits _{1\le i \le n}} \frac{\left( \sum _{i=1}^n (r_i+s_i)\right) ^{-1}}{r_1!s_1!\cdots r_n!s_n!} [ Z^{r_1} W^{s_1} \ldots Z^{r_n} W^{s_n} ] \end{aligned}$$

(9.1)

where $[Z^{r_1} W^{s_1} \cdots Z^{r_n} W^{s_n} ]$ is equal to

$$\begin{aligned}{}[ \underbrace{Z,\ldots [Z}_{r_1} , \ldots [ \underbrace{W,\ldots [W}_{s_1}, \ldots , [ \underbrace{Z,[Z,\ldots [Z}_{r_n} ,[ \underbrace{W,[W,\ldots W}_{s_n} ]]\ldots ]]. \end{aligned}$$

The Baker–Cambpell–Hausdorff theorem says that such a universal formula exists and that $\log (\hbox {exp}(Z) \hbox {exp}(W)) - (Z+W)$ is a bracket polynomial in $Z$ and $W$. Moreover if $G$ is a simply connected nilpotent Lie group then the exponential map $\hbox {exp}: {\mathfrak g} \rightarrow G$ is an analytic diffeomorphism (cf. [5, Theorem 1.2.1]). In such a case the sum defining Eq. (9.1) contains finitely many non-zero terms.

Let us explain how to apply these ideas to the study of the elements of the group $\hbox {Diff}({\mathbb R}^{2},0)$ of real analytic germs of diffeomorphism defined in a neighborhood of $(0,0)$. Any element $\varphi \in \hbox {Diff}({\mathbb R}^{2},0)$ has a power series expansion of the form

$$\begin{aligned} \varphi (x,y) = \left( \sum _{j+k \ge 1} a_{j,k} x^{j} y^{k}, \sum _{j+k \ge 1} b_{j,k} x^{j} y^{k} \right) \end{aligned}$$

where $\sum _{j+k \ge 1} a_{j,k} x^{j} y^{k}$ and $\sum _{j+k \ge 1} b_{j,k} x^{j} y^{k}$ are convergent power series with real coefficients and $(a_{1,0} x + a_{0,1} y , b_{1,0} x + b_{0,1} y)$ is a linear isomorphism. In fact the previous conditions on $\sum _{j+k \ge 1} a_{j,k} x^{j} y^{k}$ and $\sum _{j+k \ge 1} b_{j,k} x^{j} y^{k}$ determine a unique element of $\hbox {Diff}({\mathbb R}^{2},0)$ by the inverse function theorem. Let ${\mathfrak m}$ be the maximal ideal of the local ring ${\mathbb R}\{x,y\}$. We define the order $\nu (\varphi )$ of contact with the identity as

$$\begin{aligned} \nu (\phi ) = \max \bigg \{ l \in {\mathbb Z}^{+}; \sum _{j+k \ge 1} a_{j,k} x^{j} y^{k} - x \in {\mathfrak m}^{l} \quad \text {and} \quad \sum _{j+k \ge 1} b_{j,k} x^{j} y^{k} - y \in {\mathfrak m}^{l} \bigg \}. \end{aligned}$$

We define the group $\hbox {Diff}_{1}({\mathbb R}^{2},0)=\{ \varphi \in \hbox {Diff}({\mathbb R}^{2},0); \ \nu (\varphi ) \ge 2\}$ of tangent to the identity elements. Consider the group $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ of $k$-jets of tangent to the identity elements for $k \in {\mathbb Z}^{+}$. It is the subgroup of $\hbox {GL}({\mathfrak m}/{\mathfrak m}^{k+1}, {\mathbb R})$ defined by the action by composition of $\hbox {Diff}_{1}({\mathbb R}^{2},0)$ in ${\mathfrak m}/{\mathfrak m}^{k+1}$ where a map $\varphi \in \hbox {Diff}_{1}({\mathbb R}^{2},0)$ induces a linear map

$$\begin{aligned} \begin{array}{ccc} {\mathfrak m}/{\mathfrak m}^{k+1} &{} \longrightarrow &{} {\mathfrak m}/{\mathfrak m}^{k+1} \\ g + {\mathfrak m}^{k+1} &{} \longmapsto &{} g \circ \varphi + {\mathfrak m}^{k+1}. \end{array} \end{aligned}$$

The group $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ is a contractible matrix algebraic Lie group composed of unipotent elements for any $k \in {\mathbb Z}^{+}$. It is nilpotent since $\nu ([\varphi , \eta ]) > \max \{ \nu (\varphi ), \nu (\eta )\}$ for all $\varphi , \eta \in \hbox {Diff}_{1}({\mathbb R}^{2},0)$. Thus we can apply Eq. (9.1) in $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ and it extends to $\hbox {Diff}_{1}({\mathbb R}^{2},0)$ since $\hbox {Diff}_{1}({\mathbb R}^{2},0)$ is contained in the projective limit $\lim _{\leftarrow } j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$. There is a subtility in this construction since in general Formula (9.1) does not define a convergent power series and it is then necessary to define the exponential of a formal vector field via Taylor’s formula

$$\begin{aligned} \hbox {exp} (Z) = (x \circ \hbox {exp} (Z), y \circ \hbox {exp} (Z)) = \left( x + \sum _{j=1}^{\infty } \frac{Z^{j} (x)}{j!} , y + \sum _{j=1}^{\infty } \frac{Z^{j} (y)}{j!} \right) \end{aligned}$$

(9.2)

where $Z$ is understood as an operator on functions and $Z^{j}$ is the $j$th iterate of $Z$. The convergence of the Baker–Campbell–Hausdorff formula is not a problem in the following since in all subsequent applications of Formula (9.1) the sum is finite.

Proof of Theorem 9.2

Consider $l=\sigma$ in the construction of ${\mathcal G}$. We define $G = \hbox {exp}({\mathcal G})$. It is a subset of $\hbox {Diff}_{+}^{\omega }({\mathbb S}^{2})$ by compactness of ${\mathbb S}^{2}$. Since $0 \in \hbox {Fix}(G)$ and the vanishing order of any vector field in ${\mathcal G}$ at $(0,0)$ is higher than $2$ then we can consider $G$ as a subset of $\{ \varphi \in \hbox {Diff}_{1}({\mathbb R}^{2},0); \ \nu (\varphi ) \ge 3 \}$ by Eq. (9.2). The use of Eq. (9.1) implies that since ${\mathcal G}$ is a Lie algebra then $G$ is a group. In particular $G$ is a subgroup of both $\hbox {Diff}_{1}({\mathbb R}^{2},0)$ and the group of real analytic diffeomorphisms of the sphere. The torsion-free nature of $G$ is a consequence of the analogous property for $\hbox {Diff}_{1}({\mathbb R}^{2},0)$ (indeed $\nu (\phi ) = \nu (\phi ^{k})$ for all $\phi \in \hbox {Diff}_{1}({\mathbb R}^{2},0) - \{Id\}$ and $k \in {\mathbb Z}^{*}$).

There are no non-trivial elements in $G$ with arbitrarily high order of contact with the identity at $(0,0)$; otherwise the analogous property is satisfied for ${\mathcal G}$ by Eq. (9.2) and this is impossible since ${\mathcal G}$ is finite dimensional. Therefore $G$ can be interpreted as a subgroup of $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ for some $k \in {\mathbb Z}^{+}$ big enough. There exists an equivalence of categories between the finite dimensional nilpotent Lie algebras and the unipotent affine algebraic groups over fields of characteristic $0$ (cf. [6][IV, § 2, nº 4, Corollaire 4.5]). Therefore the group $j^{k} G$ induced by $G$ in $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ is a connected affine algebraic group for any $k \in {\mathbb Z}^{+}$. As a consequence $G$ is isomorphic to a connected affine algebraic group whose Lie algebra is isomorphic to ${\mathcal G}$. In this context the nilpotency class of the group and its Lie algebra coincide (cf. [6][IV, § 4, nº 1, Corollaire 1.6]). We deduce that $G$ is a $\sigma$-step nilpotent group.$\square$

Remark 9.4

It is easy to show by hand that $\langle X, \ldots , \alpha ^{l-j-1} X \rangle _{\mathbb R}$ is the Lie algebra of $G_{(j)}$ for any $0 < j <l$ and $G_{(l)}=\{Id\}$.

Proof of Theorem 9.3

Consider $l=\sigma$ in the construction of ${\mathcal G}$. We define the group

$$\begin{aligned} J = \langle \hbox {exp}(t X), \hbox {exp}(t \alpha X), \ldots , \hbox {exp}(t \alpha ^{\sigma -1} X), \hbox {exp}(t Y) \rangle \end{aligned}$$

for some fixed $t \in {\mathbb R}^{*}$ small. Analogously as in the Proof of Theorem 9.2 we can consider that $G$ and $J$ are subgroups of $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ for some $k \in {\mathbb Z^{+}}$ big enough. Let $H$ be the smallest algebraic group in $j^{k} \hbox {Diff}_{1}({\mathbb R}^{2},0)$ containing $J$. The group $H$ is a connected unipotent algebraic group contained in $G$. The Lie algebra ${\mathfrak h}$ of $H$ coincides with ${\mathcal G}$ by construction. As a consequence the groups $G$ and $H$ also coincide. The group $J$ is $\sigma$-step nilpotent since its algebraic closure is. We can obtain $\sigma$-step nilpotent subgroups of $\hbox {Diff}_{+}^{\omega }({\mathbb R}^{2},0)$ sharing $(0,0)$ as a fixed point and with generators arbitrarily and uniformly close to the identity map by Proposition 9.5 below.$\square$

The next result completes the Proof of Theorem 9.3.

Proposition 9.5

Let $Z \in {\mathcal G}$. The diffeomorphism $\hbox {exp}(t Z)$ converges to the identity map in the strong $C^{1}$-topology when $t \rightarrow 0$.

Proof

The result is obviously true for the $C^{1}$-topology for diffeomorphisms of the sphere. We will show that the result still holds true in $\hbox {Diff}^{1}({\mathbb R}^{2},0)$.

Let $\eta _{t} = \hbox {exp}(t Z)$. The diffeomorphism $\eta _{t}$ converges to the identity map in any compact set of the plane. Let us study the properties of the one parameter flow $\eta _{t}$ in the neighborhood of $\infty$. Let $z$ be a complex coordinate in the Riemann sphere. We consider the coordinate $w=1/z$ in order to study the behavior of the diffeomorphisms in the neighborhood of the point $z=\infty$. The vector field $\partial /\partial z$ is equal to $-w^{2} \partial /\partial w$. We obtain

$$\begin{aligned} \frac{\partial }{\partial x} = - (\hat{x}^{2} - \hat{y}^{2}) \frac{\partial }{\partial \hat{x}} - 2 \hat{x} \hat{y} \frac{\partial }{\partial \hat{y}} \quad \text {and} \quad \frac{\partial }{\partial y} = 2 \hat{x} \hat{y} \frac{\partial }{\partial \hat{x}} - (\hat{x}^{2} - \hat{y}^{2}) \frac{\partial }{\partial \hat{y}} \end{aligned}$$

where $z=x+i y$ and $w = \hat{x} + i \hat{y}$. The vanishing order of $Z$ at $\infty$ is higher than $2$. The expression $\hat{\eta }_{t} = (1/z) \circ \eta _{t} \circ (1/w)$ of the diffeomorphism $\eta _{t}$ in the $w$-coordinate satisfies

$$\begin{aligned} \hat{\eta }_{t}(w) = w + t \! \! \sum _{j+k \ge 3} c_{j,k}(t) \hat{x}^{j} \hat{y}^{k} \end{aligned}$$

in the coordenate $w$ where $c_{j,k}$ is a polynomial with complex coefficients for $j + k \ge 3$. Since

$$\begin{aligned} \eta _{t}(z)- z = \eta _{t} \left( \frac{1}{w} \right) - \frac{1}{w} = \frac{1}{\hat{\eta }_{t}(w)} - \frac{1}{w} = \frac{1}{w + t O(w^{3}) } - \frac{1}{w} = O(t w) \end{aligned}$$

then $\eta _{t}$ converges to the identity map in the neighborhood of $\infty$ in the $C^{0}$-topology. The expression

$$\begin{aligned} \frac{\partial }{\partial x} (\eta _{t}(z) - z)&= \left( - (\hat{x}^{2} - \hat{y}^{2}) \frac{\partial }{\partial \hat{x}} - 2 \hat{x} \hat{y} \frac{\partial }{\partial \hat{y}}\right) \left( \eta _{t} \left( \frac{1}{w} \right) - \frac{1}{w} \right) \\&= \left( - (\hat{x}^{2} - \hat{y}^{2}) \frac{\partial }{\partial \hat{x}} - 2 \hat{x} \hat{y} \frac{\partial }{\partial \hat{y}}\right) \left( \frac{1}{\hat{\eta }_{t}(w)} - \frac{1}{w} \right) = O(t w^{2}) \end{aligned}$$

and the analogue for $(\partial /\partial y)(\eta _{t}(z) - z)$ imply that $\hbox {exp}(tZ)$ converges to $Id$ in the neighborhood of $\infty$ in the $C^{1}$-topology when $t \rightarrow 0$.$\square$

Remark 9.6

It is easy to check out that $\lim _{t \rightarrow 0} \hbox {exp}(t Z) = Id$ in the strong $C^{k}$-topology for any $k \in {\mathbb Z}^{+}$.