Bounded limit cycles of polynomial foliations of \mathbb{C}P^{2}

Bounded limit cycles of polynomial foliations of \mathbb{C}P^{2}

Nataliya Goncharuk The research was supported by Russian President grant No. MD-2859.2014.1 and by Simons Foundation.Research of both authors was supported by RFBR project 13-01-00969-a, and UNAM-DGAPA-PAPIIT Projects IN 103914 and IN 102413.Address (both authors): National Research University Higher School of Economics, Russia, Moscow, Myasnitskaya str., 20    Yury Kudryashov This research carried out in 2014–2015 was supported by “The National Research University ‘Higher School of Economics’ Academic Fund Program” grant No. 14-01-0193.
July 13, 2019 \currenttime
Abstract

In this article we prove in a new way that a generic polynomial vector field in \mathbb{C}^{2} possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.

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[1]Contentscontents

1 Introduction

Consider a polynomial differential equation in \mathbb{C}^{2} (with complex time),

\displaystyle{\dot{x}} \displaystyle=P(x,y), (1)
\displaystyle{\dot{y}} \displaystyle=Q(x,y),

where \max(\deg P,\deg Q)=n. The splitting of \mathbb{C}^{2} into trajectories of this vector field defines a singular analytic foliation of \mathbb{C}^{2}. Denote by \mathcal{A}_{n} the space of foliations of \mathbb{C}^{2} defined by vector fields (1) of degree at most n with coprime P and Q. Two vector fields define the same foliation if they are proportional, hence \mathcal{A}_{n} is a Zariski open subset of the projective space. \mathcal{A}_{n} is equipped with a natural topology induced from this projective space.

In 1970-s there appeared several results on the properties of generic foliations from \mathcal{A}_{n}. In particular, Yu. Ilyashenko [Il78] proved that a generic foliation (more precisely, each foliation from some full Lebesgue measure subset of \mathcal{A}_{n}) has an infinite number of limit cycles. Later his theorem was improved by E. Rosales-González, L. Ortiz-Bobadilla and A. Shcherbakov [SRO98], namely they replaced a full-measure set by an open dense subset.

Definition.

Limit cycle on a leaf L of an analytic foliation is an element [\gamma] of the free homotopy group of L such that the holonomy along (any) its representative \gamma is non-identical.

Definition.

A set of limit cycles of a foliation is called homologically independent, if for any leaf L all the cycles located on this leaf are linearly independent in H_{1}(L).

Theorem ([Il78]).

For n\geqslant 2, there exists a full-measure subset of \mathcal{A}_{n}, such that each foliation from this subset possesses an infinite number of homologically independent limit cycles.

Theorem ([SRO98]).

For n\geqslant 3, there exists an open dense subset of \mathcal{A}_{n}, such that each foliation from this subset possesses an infinite number of homologically independent limit cycles.

The proof of the first theorem in [Il78] is rather technical; the proof of the second one in [SRO98] contains about 10 pages of cumbersome estimates of integrals along the limit cycles. The constructed sequence of representatives \gamma_{j} of required limit cycles [\gamma_{j}] in both theorems converges to the infinite line.

Our results yield another, less technical proof of these theorems, and our limit cycles are detached from the infinite line. Also, our proof works for n=2 for both types of genericity assumptions.

Main Theorem.

For n\geqslant 2, there exist

  • a full-measure subset \mathcal{A}^{LC1}_{n}\subset\mathcal{A}_{n},

  • a complement to a real-analytic subset \mathcal{A}^{LC2}_{n}\subset\mathcal{A}_{n},

such that each \mathcal{F}\in\mathcal{A}_{n}^{LC1}\cup\mathcal{A}_{n}^{LC2} possesses an infinite sequence of limit cycles [\gamma_{j}] such that:

  • the cycles are homologically independent;

  • the multipliers of the cycles tend to zero;

  • the cycles are uniformly bounded, i.e., there exists a ball in \mathbb{C}^{2} that includes all representatives \gamma_{j};

  • there exists a cross-section such that \gamma_{j} intersect it in a dense subset.

The explicit descriptions of the sets \mathcal{A}^{LC1}_{n} and \mathcal{A}^{LC2}_{n} are given below, in Sections “Multiplicative density” and “Unsolvable monodromy group”, respectively.

The key genericity assumption for \mathcal{A}^{LC1}_{n} is that the characteristic numbers of two singular points at infinity generate a dense semi-group in \mathbb{C}/\mathbb{Z}. The key genericity assumption for \mathcal{A}^{LC2}_{n} is that the monodromy group at infinity is unsolvable.

Though the exceptional set in the second part is much thinner, we include the first part for two reasons: \mathcal{A}^{LC2}_{n} does not include \mathcal{A}^{LC1}_{n}, and the first case is technically easier.

Our construction also yields that the infinite number of limit cycles survives in a neighbourhood of \mathcal{A}^{LC1}_{n}\cup\mathcal{A}^{LC2}_{n} in the space \mathcal{B}_{n+1} of foliations of \mathbb{C}P^{2} that are given by a polynomial vector field of degree at most n+1 in any affine chart, see Corollary 6.

2 Preliminaries

2.1 Extension to infinity

Let us extend a polynomial foliation \mathcal{F}\in\mathcal{A}_{n} given by (1) to \mathbb{C}P^{2}. After changing variables, u=\frac{1}{x},v=\frac{y}{x}, and the time change d\tau=-u^{n-1}dt, the vector field takes the form

\displaystyle{\dot{u}} \displaystyle=u{\widetilde{P}}(u,v) (2)
\displaystyle{\dot{v}} \displaystyle=v{\widetilde{P}}(u,v)-{\widetilde{Q}}(u,v)

where {\widetilde{P}}(u,v)=P\left(\frac{1}{u},\frac{v}{u}\right)u^{n} and {\widetilde{Q}}(u,v)=Q\left(\frac{1}{u},\frac{v}{u}\right)u^{n} are two polynomials of degree at most n.

Since {\dot{u}}(0,v)\equiv 0, the infinite line \set{u=0} is invariant under this vector field. Denote by h(v) the polynomial {\dot{v}}(0,v)=v{\widetilde{P}}(0,v)-{\widetilde{Q}}(0,v). In a generic (non-dicritical) case h(v)\not\equiv 0; then (2) has isolated singular points a_{j}\in\set{u=0} at the roots of h, and L_{\infty}≔\set{u=0}\smallsetminus\{a_{1},a_{2},\ldots\} is a leaf of the extension of \mathcal{F} to \mathbb{C}P^{2}.

Denote by \mathcal{A}_{n}^{\prime} the set of foliations \mathcal{F}\in\mathcal{A}_{n} such that h has n+1 distinct roots a_{j}, j=1,\ldots,n+1. In particular, all these foliations are non-dicritical.

For each j, let \lambda_{j} be the ratio of the eigenvalues of the linearization of (2) at a_{j} (the eigenvalue corresponding to L_{\infty} is in the denominator). One can show that \sum\lambda_{j}=1, and this is the only relation on \lambda_{j}.

For \mathcal{F}\in\mathcal{A}_{n}^{\prime}, fix a non-singular point O\in L_{\infty} and a cross-section S at O given by v=\operatorname{const}. Let \Omega L_{\infty} be the loop space of (L_{\infty},O), i. e., the space of all continuous maps (S^{1},pt)\rightarrow(L_{\infty},O). For a loop \gamma\in\Omega L_{\infty}, denote by \mathbf{M}_{\gamma}:(S,O)\rightarrow(S,O) the monodromy map along \gamma. It is easy to see that \mathbf{M}_{\gamma} depends only on the class [\gamma]\in\pi_{1}(L_{\infty},O), and the map \gamma\mapsto\mathbf{M}_{\gamma} reverses the order of multiplication,

\mathbf{M}_{\gamma\gamma^{\prime}}=\mathbf{M}_{\gamma^{\prime}}\circ\mathbf{M}% _{\gamma}.

The set of all possible monodromy maps \mathbf{M}_{\gamma}, \gamma\in\Omega L_{\infty}, is called the monodromy pseudogroup G=G(\mathcal{F}). The word “pseudogroup” means that there is no common domain where all elements of G are defined. However we will follow the tradition and write “monodromy group” instead of “monodromy pseudogroup”.

Choose n+1 loops \gamma_{j}\in\Omega L_{\infty}, j=1,2,\ldots,n+1, passing around points a_{j}, respectively. We suppose that \gamma_{j} are simple and intersect only at O. Then the pseudogroup G(\mathcal{F}) is generated by monodromy maps \mathbf{M}_{j}=\mathbf{M}_{\gamma_{j}}. It is easy to see that the multipliers \mu_{j}=\mathbf{M}_{j}^{\prime}(0) are equal to \exp{2\pi i\lambda_{j}}.

2.2 Fatou coordinates

The space of germs of analytic parabolic maps g:(\mathbb{C},0)\rightarrow(\mathbb{C},0), z\mapsto z+o(z), has a natural filtration by the degree of the leading term of g(z)-z. Denote by A_{p} the set of germs of the form z\mapsto z+az^{p+1}+o(z^{p+1}), a\neq 0.

In this section we will recall some results on sectorial rectifying charts of parabolic fixed points that will be used in the article. For a more complete exposition, see, e.g., Chapter IV of [IYa07].

We start with describing the formal normal forms for quadratic parabolic germs.

Theorem.

A quadratic parabolic germ f:z\mapsto z+az^{2}+bz^{3}+o(z^{3}) is formally conjugated to the time-one flow map f_{\lambda} of the vector field v_{\lambda}(z)=\frac{z^{2}}{1+\lambda z}, where \lambda=1-\frac{b}{a^{2}}. More precisely, there exists a formal series H(z)=az+\sum_{k=2}^{\infty}h_{k}z^{k}, such that f_{\lambda}\circ H=H\circ f. The series H is uniquely defined modulo a formal composition with a flow map of v_{\lambda}.

Remark.

It is easy to see that the map t_{\lambda}:z\mapsto-\frac{1}{z}+\lambda\log z conjugates f_{\lambda} to the map t\mapsto t+1, t_{\lambda}(f_{\lambda}(z))=t_{\lambda}(z)+1.

We will need the following theorem that describes sectorial rectifying charts for quadratic parabolic germs. Consider the following sectors

\displaystyle S_{\alpha,r}^{+} \displaystyle=\Set{z}{|z|<r,|\arg z|<\alpha}, \displaystyle S_{\alpha,r}^{-} \displaystyle=\Set{z}{|z|<r,|\arg z-\pi|<\alpha}.
Theorem (Sectorial Normalization Theorem).

Let f:z\mapsto z+az^{2}+o(z^{2}) be a quadratic parabolic map, let H(z)=az+\sum_{k=2}^{\infty}h_{k}z^{k} be a formal series which conjugates f to its formal normal form f_{\lambda}. Then for any \frac{\pi}{2}<\alpha<\pi there exists r>0 and a unique couple of analytic maps h^{\pm}:\frac{1}{a}S^{\pm}_{\alpha,r}\rightarrow\mathbb{C} with the following properties:

  • H is an asymptotic series for h^{-} and h^{+}: for N\in\mathbb{N}, we have h^{\pm}(z)=az+\sum_{k=2}^{N}h_{k}z^{k}+o(z^{N}) as z\rightarrow 0 inside \frac{1}{a}S^{\pm}_{\alpha,r};

  • h^{\pm} conjugates f to f_{\lambda}: f_{\lambda}\circ h^{\pm}=h^{\pm}\circ f.

Remark.

For most parabolic germs f, h^{-}\neq h^{+}. So, the analytic classification of parabolic germs does not coincide with their formal classification. The analytic classification has functional moduli called Ecalle–Voronin moduli, namely the restrictions of (h^{+})^{-1}\circ h^{-} to the sectors \set{z}{|z|<r,\pi-\alpha<\arg z<\alpha} and \set{z}{|z|<r,-\alpha<\arg z<\pi+\alpha} up to a conjugation by a flow map of v_{\lambda}.

Remark.

It is easy to check that the image of h^{\pm} includes a sector of the form S^{\pm}_{\alpha^{\prime},r^{\prime}} for each \alpha^{\prime}<\alpha and some r^{\prime}, and is included by another sector of the same form. Also, t_{\lambda}(S^{-}_{\alpha^{\prime},r^{\prime}}) includes a sector at infinity:

S_{\beta,R}^{\infty}=\Set{\zeta}{|\zeta|>R,|\arg\zeta|<\beta} (3)

for each \beta<\alpha^{\prime} and some R=R(\alpha^{\prime},r^{\prime},\lambda,\beta)\gg 1. Thus the image of \zeta=t_{\lambda}\circ h^{-} includes a sector at infinity.

Definition.

A Fatou coordinate for a parabolic map f in a sector \frac{1}{a}S^{-}_{\alpha,r} is a coordinate of the form \zeta=t_{\lambda}\circ h^{-}, where h^{-} is given by Sectorial Normalization Theorem. A Fatou coordinate \zeta conjugates f to the shift \zeta\mapsto\zeta+1 in a domain that includes a sector at infinity (3), and is defined uniquely modulo addition of a complex number.

We shall need the following statement.

Lemma 1.

Let g be a parabolic map of the form z\mapsto z+az^{2}+\ldots. Let \zeta be a Fatou chart for g defined in a sector \frac{1}{a}S^{-}_{\alpha,r}. Let S^{\infty} be the image of a smaller sector \frac{1}{a}S^{-}_{\alpha-\varepsilon,r-\varepsilon} under \zeta. Let F:\mathbb{C}\rightarrow\mathbb{C} be an analytic map, F(0)=0, defined in the chart z. Let {\tilde{F}}=\zeta\circ F\circ\zeta^{-1} be the corresponding map in the chart \zeta.

  • If F(z)=kz+o(z), then {\tilde{F}}(\zeta)=k^{-1}\zeta+c+o(1) as \zeta\rightarrow\infty inside S^{\infty}.

  • If F(z)=z+kz^{p+1}+o(z^{p+1}), p\geqslant 1, then {\tilde{F}}(\zeta)=\zeta+(-1)^{p-1}ka^{-p}\frac{1}{\zeta^{p-1}}+o\left(\frac{1% }{\zeta^{p-1}}\right) in \zeta-chart as \zeta\rightarrow\infty inside S^{\infty}.

  • If F is a parabolic map, then \log{\tilde{F}}^{\prime}(\zeta)=o(F(\zeta)-\zeta) as \zeta\rightarrow\infty inside S^{\infty}.

Proof.

Recall that h(z)-az=O(z^{2}), z\in\frac{1}{a}S^{-}_{\alpha,r}. The Cauchy estimates imply that h^{\prime}(z)=a+O(z) in \frac{1}{a}S_{\alpha-\varepsilon,r-\varepsilon}.

Let us prove b). Note that

\displaystyle F(h^{-1}(w)) \displaystyle=h^{-1}(w)+k(h^{-1}(w))^{p+1}+o((h^{-1}(w))^{p+1})
\displaystyle=h^{-1}(w)+ka^{-p-1}w^{p+1}+o(w^{p+1}),

hence

(h\circ F\circ h^{-1})(w)=w+\int_{h^{-1}(w)}^{F(h^{-1}(w))}a+O(z)\,dz=w+ka^{-p% }w^{p+1}+o(w^{p+1}).

Similarly, for \zeta=t_{\lambda}(w) we have

\displaystyle{\tilde{F}}(\zeta) \displaystyle=\zeta+\int_{w}^{w+ka^{-p}w^{p+1}+o(w^{p+1})}t_{\lambda}^{\prime}% (\omega)\,d\omega
\displaystyle=\zeta+\int_{w}^{w+ka^{-p}w^{p+1}+o(w^{p+1})}\frac{1}{\omega^{2}}% +\frac{\lambda}{\omega}\,d\omega
\displaystyle=\zeta+ka^{-p}w^{p-1}+o(w^{p-1})
\displaystyle=\zeta+(-1)^{p-1}ka^{-p}\frac{1}{\zeta^{p-1}}+o\left(\frac{1}{% \zeta^{p-1}}\right)

Assertion a) can be proved in the same way. Finally, the last assertion follows from Assertion b) in \zeta\left(\frac{1}{a}S^{-}_{\alpha-\frac{\varepsilon}{2},r-\frac{\varepsilon}% {2}}\right) and Cauchy estimates. ∎

2.3 Unsolvability of the monodromy group

In [Shch84], A. Shcherbakov proved that for a generic foliation \mathcal{F}\in\mathcal{A}_{n}, the monodromy group is unsolvable. It turns out that a group of germs (\mathbb{C},0)\rightarrow(\mathbb{C},0) is unsolvable if and only if it contains a couple of commutators that do not commute with each other. This follows from the next lemma.

Lemma 2.

Let f(z)=z+az^{p+1}+o(z^{p+1}) and g(z)=z+bz^{q+1}+o(z^{q+1}) be two parabolic germs. Then

[f,g](z)≔(f\circ g\circ f^{-1}\circ g^{-1})(z)=z+ab(p-q)z^{p+q+1}+o(z^{p+q+1}).

In particular, if p\neq q, a\neq 0, b\neq 0, then [f,g]\in A_{p+q}.

Corollary.

If a group G of germs (\mathbb{C},0)\rightarrow(\mathbb{C},0) contains two parabolic germs g_{1}\in A_{p}, g_{2}\in A_{q} with p\neq q, then G is unsolvable.

Proof.

Indeed, none of the commutators g_{3}=[g_{1},g_{2}]\in A_{p+q}, g_{4}=[g_{3},g_{2}]\in A_{p+2q}, … can be the identity map. ∎

The main result of this section is the following lemma.

Lemma 3.

There exists an open dense subset \mathcal{A}_{n} such that for each foliation from this subset

\forall i\neq j\quad[\mathbf{M}_{i},\mathbf{M}_{j}]\in A_{1},\quad[\mathbf{M}_% {i}^{-1},\mathbf{M}_{j}]\in A_{1},\quad[[\mathbf{M}_{i},\mathbf{M}_{j}],[% \mathbf{M}_{i}^{-1},\mathbf{M}_{j}]]\in A_{3}.

This lemma is immediately implied by the following two statements.

Lemma 4.

There exists a real analytic subset \mathcal{E}\subset\mathcal{A}_{n} of positive codimension such that for \mathcal{F}\notin\mathcal{E}

  • all commutators [\mathbf{M}_{i},\mathbf{M}_{j}], i\neq j, belong to A_{1};

  • all numbers \dfrac{S(\mathbf{M}_{i})(0)}{\mathbf{M}_{i}^{\prime}(0)^{2}-1}, i=1,\ldots,n+1, are different.

Here and below S(f) is the Schwarzian derivative of f,

S(f)(z)=\frac{f^{\prime\prime\prime}(z)}{f^{\prime}(z)}-\frac{3}{2}\left(\frac% {f^{\prime\prime}(z)}{f^{\prime}(z)}\right)^{2}.

In [Shch84], Shcherbakov proved that for a generic foliation, at least one commutator [\mathbf{M}_{i},\mathbf{M}_{j}] belongs to A_{1}, see Section 6.3 of [Shch06]. But it is easy to extend this result to all pairs i\neq j using analytic continuation along loops in \mathcal{A}_{n} that permute the singular points. The second part is proved in the same article but not explicitly stated, so one needs to go through the proof of Theorem 9 in [Shch06] to verify that the assertion of the corollary after Lemma 5 is the only property of \mathcal{F} used in the proof.

Similar results were achieved in [BLL97, N94].

Lemma.

Consider two hyperbolic germs f,g such that

  • f^{\prime}(0)^{2}\neq 1, g^{\prime}(0)^{2}\neq 1;

  • [f,g]\in A_{1}, i.e., [f,g]^{\prime\prime}(0)\neq 0;

  • \displaystyle\frac{S(f)(0)}{f^{\prime}(0)^{2}-1}\neq\frac{S(g)(0)}{g^{\prime}(% 0)^{2}-1}.

Then [f,g] does not commute with [f^{-1},g]; moreover, [[f,g],[f^{-1},g]]\in A_{3}.

This lemma is motivated by Proposition 7 in [Shch84] (which coincides with the corollary after Lemma 5 in [Shch06]) but provides an explicit pair of commutators that do not commute.

Proof.

One can verify that

S([f,g])(0)=\left[\left(\frac{S(f)}{(f^{\prime})^{2}-1}-\frac{S(g)}{(g^{\prime% })^{2}-1}\right)\left(1-\frac{1}{(f^{\prime})^{2}}\right)\left(1-\frac{1}{(g^{% \prime})^{2}}\right)\right]_{z=0}

thus

\frac{S([f^{-1},g])(0)}{S([f,g])(0)}=-f^{\prime}(0)^{2}

On the other hand,

\frac{[f^{-1},g]^{\prime\prime}(0)}{[f,g]^{\prime\prime}(0)}=-f^{\prime}(0). (4)

Two last equalities prove that

\frac{S([f^{-1},g](0)}{[f^{-1},g]^{\prime\prime}(0)}=f^{\prime}(0)\frac{S([f,g% ](0)}{[f,g]^{\prime\prime}(0)}

The assertions of the lemma imply that this is not zero, thus

\frac{S([f^{-1},g](0)}{[f^{-1},g]^{\prime\prime}(0)}\neq\frac{S([f,g](0)}{[f,g% ]^{\prime\prime}(0)}.

Finally, expanding [f,g]\circ[f^{-1},g]-[f^{-1},g]\circ[f,g] up to the fourth order, one can check that the above inequality is equivalent to

([f,g]\circ[f^{-1},g]-[f^{-1},g]\circ[f,g])^{(4)}(0)\neq 0,

hence [[f,g],[f^{-1},g]]\in A_{3}. ∎

3 Plan of the proof of Main Theorem

We will construct the limit cycles as the lifts of loops in the infinite line. Note that if the monodromy map \mathbf{M}_{k_{1}}\mathbf{M}_{k_{2}}\ldots\mathbf{M}_{k_{l}}:S\rightarrow S has a fixed point p\neq 0, then the corresponding loop \gamma=\gamma_{k_{l}}\gamma_{k_{l-1}}\ldots\gamma_{k_{1}} lifts to a limit cycle c starting from p; the projection of c to the infinite line is \gamma.

We proceed in two steps. First we construct contracting monodromy maps that satisfy inclusion, contraction and covering assumptions formulated below. This is done in a different way for two types of genericity assumptions, see Section “Construction of contracting maps”. Then we use the maps constructed on the first step to obtain limit cycles that satisfy the assertions of Main Theorem. On this step we use no information about the foliation except for existence of maps with prescribed properties, see Section “Construction of limit cycles”.

3.1 Step 1: contracting maps

We shall find two topological discs \Delta^{-}\subset\Delta^{+}\subset S in the cross-section, 0\notin\Delta^{+}, an analytic chart \zeta in \Delta^{+} and a tuple of monodromy maps f_{j} with the following properties. Each f_{j} is a composition of standard generators \mathbf{M}_{k} of the monodromy group at infinity. For any splitting of this composition into two parts f_{j}=f_{j}^{(t)}\circ f_{j}^{(h)}, we will say that f_{j}^{(t)} is a tail of f_{j} and f_{j}^{(h)} is its head.

Inclusion:

f_{j}(\Delta^{+})\subset\Delta^{+} for any j.

Contraction:

All compositions of the form f_{i}^{(t)}\circ f_{j}^{(h)}, f_{i}^{(h)}\neq\operatorname{id}, f_{j}^{(h)}\neq\operatorname{id}, contract in (f_{j}^{(h)})^{-1}\circ f_{i}^{(h)}(\Delta^{+})\cap\Delta^{+} with respect to the chart \zeta. In particular, all f_{j} contract in \Delta^{+}.

Covering:

Images of \Delta^{-} under f_{j} cover \Delta^{-}.

We will also suppose that the compositions f_{j} do not contain identical subcompositions, otherwise we remove them. Obviously this does not break any of other requirements on f_{j}.

3.2 Step 2: limit cycles

Here we use the maps f_{j} to construct infinitely many homologically independent limit cycles. We will use not a particular construction of f_{j}, but only the assumptions inclusion, contraction and covering, so the arguments work for both sets A^{LC1}_{n},A^{LC2}_{n}. The main motivation is the following lemma1. \raisebox{10.0pt}{\hypertarget{id17}{}}\hyperlink{id16}{1}\raisebox{10.0pt}{\hypertarget{id17}{}}\hyperlink{id16}{1}footnotetext: Some people attribute this statement to Hutchinson [H81], but we failed to find exactly this statement in this article.

Lemma 5.

Under assumptions above, for an open U\subset\Delta^{-} and \varepsilon>0 there exists a word w=j_{1}\ldots j_{N} such that the monodromy map f_{w}=f_{j_{1}}\circ f_{j_{2}}\circ\ldots\circ f_{j_{N}} satisfies f_{w}(\Delta^{+})\subset U and |f^{\prime}_{w}|<\varepsilon in \Delta^{+}.

Remark.

We will use this lemma only for \varepsilon<1. In this case, the map f_{w} obviously has a fixed point in U. It corresponds to a limit cycle with arbitrarily small multiplier which passes through U.

This lemma enables us to prove assertions b)–d) of Main Theorem. The proof of homological independence is more complicated.

Proof.

Take a point p\in U\subset\Delta^{-}. Due to the covering assumption, there exists an index j_{1} such that p\in f_{j_{1}}(\Delta^{-}). Now, take the preimage f_{j_{1}}^{-1}(p)\subset\Delta^{-}, and repeat the arguments; we obtain a map f_{j_{2}} such that p\in f_{j_{1}}(f_{j_{2}}(\Delta^{-})).

Repeating the procedure, we get a word w=j_{1}\,j_{2}\,\ldots\,j_{N} such that f_{w}(\Delta^{-})=f_{j_{1}}\circ f_{j_{2}}\circ\cdots\circ f_{j_{N}}(\Delta^{-}) contains p. The diameter of the image f_{w}(\Delta^{+}) tends to zero as N tends to infinity, since all maps f_{j} contract in \Delta^{+}. So, if N is large enough, the fact that p\in f_{w}(\Delta^{+}) would imply f_{w}(\Delta^{+})\subset U and |f^{\prime}_{w}|<\varepsilon in the whole \Delta^{+}. ∎

3.3 A neighborhood in \mathcal{B}_{n}

Note that assumptions inclusion, contraction and covering are robust in the following sense. Considers a foliation \mathcal{F}\in\mathcal{A}_{n} that possesses a tuple of monodromy maps that satisfy these assumptions. Then there exists a bidisc D\subset\mathbb{C}^{2} and \varepsilon>0 such that any foliation \mathcal{F}^{\prime} of D which is \varepsilon-close to \mathcal{F} in D possesses monodromy maps that satisfy these assumptions. Since Step 2 relies only on these properties, such foliation \mathcal{F}^{\prime} satisfies assertions of Main Theorem. In particular, we have the following corollary.

Corollary 6.

Any foliation from some open neighborhood \mathcal{U}, \mathcal{A}^{LC1,2}_{n}\subset\mathcal{U}\subset\mathcal{B}_{n+1}, possesses an infinite number of limit cycles satisfying assertions a)–d) of the Main theorem.

4 Construction of contracting maps

4.1 Multiplicative density

We put the following genericity assumptions on the foliation:

  • the characteristic numbers of two of the singular points at infinity (say, \lambda_{1} and \lambda_{2}) generate a dense subgroup in \mathbb{C}/\mathbb{Z};

  • the corresponding monodromy maps \mathbf{M}_{1},\mathbf{M}_{2} do not commute.

Each condition defines a full measure set. For the former condition it is clear, and for the latter one see Lemma 4.

After a holomorphic coordinate change, we may and will assume that the map \mathbf{M}_{1} is linear. If this map expands, let us replace it with its inverse. Then \operatorname{Im}\lambda_{1}>0. Let us pass to the chart \zeta=\frac{\log z}{2\pi i}, \zeta\in\mathbb{C}/\mathbb{Z}. In this chart, points with large \operatorname{Im}\zeta correspond to points z close to the origin.

Let {\tilde{\mathbf{M}}}_{1}:\zeta\mapsto\zeta+\lambda_{1} and {\tilde{\mathbf{M}}}_{2} be the maps \mathbf{M}_{1} and \mathbf{M}_{2} written in the chart \zeta. These maps are defined for sufficiently large \operatorname{Im}\zeta, and

\displaystyle{\tilde{\mathbf{M}}}_{2}(\zeta) \displaystyle=\zeta+\lambda_{2}+o(1),
\displaystyle{\tilde{\mathbf{M}}}_{2}^{\prime}(\zeta) \displaystyle=1+o(1)

as \operatorname{Im}\zeta\rightarrow\infty. Since \mathbf{M}_{1} does not commute with \mathbf{M}_{2}, the map {\tilde{\mathbf{M}}}_{2} is not linear, hence {\tilde{\mathbf{M}}}_{2}^{\prime} is not identically one. Let \Delta^{+} be a small closed disc such that either {\tilde{\mathbf{M}}}_{2} or its inverse uniformly contracts in \Delta^{+}. Without loss of generality we can and shall assume that it is {\tilde{\mathbf{M}}}_{2},

\max_{\zeta\in\Delta^{+}}|{\tilde{\mathbf{M}}}_{2}^{\prime}(\zeta)|<1. (5)

Next, let \zeta_{0} be the center of \Delta^{+}, put T=\zeta_{0}-{\tilde{\mathbf{M}}}_{2}(\zeta_{0}). Note that {\tilde{\mathbf{M}}}_{2}(\Delta^{+})+T\Subset\Delta^{+}. Choose a much smaller disc \Delta^{-}\subset\Delta^{+} with the same center,

\operatorname{diam}(\Delta^{-})<\operatorname{dist}({\tilde{\mathbf{M}}}_{2}(% \partial\Delta^{+})+T,\partial\Delta^{+}). (6)

Choose a tuple of vectors T_{j}=k_{j}\lambda_{1}+l_{j}\lambda_{2}\in\mathbb{C}/\mathbb{Z} such that

\Delta^{-}\subset\bigcup_{j}({\tilde{\mathbf{M}}}_{2}(\Delta^{-})+T_{j})% \subset\bigcup_{j}({\tilde{\mathbf{M}}}_{2}(\Delta^{+})+T_{j})\subset\Delta^{+}. (7)

Due to (6), it is enough to take T_{j} such that {\tilde{\mathbf{M}}}_{2}(\Delta^{-})+T_{j} cover \Delta^{-} and |T-T_{j}|<\operatorname{diam}(\Delta^{-}). Due to the density condition, these T_{j} can be chosen of the form T_{j}=k_{j}\lambda_{1}+l_{j}\lambda_{2}.

Now, let us choose f_{j} so that they approximate the maps T_{j}\circ{\tilde{\mathbf{M}}}_{2} in \Delta^{+}. Note that the map {\tilde{\mathbf{M}}}_{2}^{l_{j}}\circ{\tilde{\mathbf{M}}}_{1}^{k_{j}} approximates the shift \zeta\mapsto\zeta+T_{j} for large \operatorname{Im}\zeta, hence for N large enough, the map {\tilde{\mathbf{M}}}_{1}^{-N}\circ{\tilde{\mathbf{M}}}_{2}^{l_{j}}\circ{\tilde% {\mathbf{M}}}_{1}^{k_{j}+N} is very close to the shift by T_{j} in C^{1}({\tilde{\mathbf{M}}}_{2}(\Delta^{+})). Therefore, we can take the maps

f_{j}=(\mathbf{M}_{1}^{-N}\circ\mathbf{M}_{2}^{l_{j}}\circ\mathbf{M}_{1}^{k_{j% }+N})\circ\mathbf{M}_{2}.

Let {\tilde{f}}_{j} be the map f_{j} written in the chart \zeta.

For sufficiently large N, these maps satisfy inclusion and covering assumptions from the plan of the proof. Obviously, each {\tilde{f}}_{j} contracts in \Delta^{+}. Now, we have to prove contraction assumption, i.e. that for N large enough all compositions {\tilde{f}}_{i}^{(t)}\circ{\tilde{f}}_{j}^{(h)} contract. Recall that {\tilde{f}}_{j}^{(h)}\neq\operatorname{id}, {\tilde{f}}_{i}^{(t)}\neq{\tilde{f}}_{i}.

Since the map {\tilde{\mathbf{M}}}_{1}^{-N}\circ{\tilde{\mathbf{M}}}_{2}^{l_{j}}\circ{\tilde% {\mathbf{M}}}_{1}^{k_{j}+N} and its heads approximate shifts in C^{1}(\Delta^{+}), for N large enough the derivative of {\tilde{f}}_{j}^{(h)} is close to the set {\tilde{\mathbf{M}}}_{2}^{\prime}(\Delta^{+}). On the other hand, the derivative of {\tilde{f}}_{i}^{(t)} on the set {\tilde{f}}_{i}^{(h)}(\Delta^{+}) is arbitrarily close to one. Thus we can make the derivative of the composition {\tilde{f}}_{i}^{(t)}\circ{\tilde{f}}_{j}^{(h)} on the set \Delta^{+}\cap\left(({\tilde{f}}_{j}^{(h)})^{-1}\circ{\tilde{f}}_{i}^{(h)}(% \Delta^{+})\right) arbitrarily close to {\tilde{\mathbf{M}}}_{2}^{\prime}(\Delta^{+}). Now (5) yields contraction assumption.

4.2 Unsolvable monodromy group

In this case, the construction is similar, but instead of the logarithmic chart we use the Fatou chart for one of the parabolic monodromy maps, and there are more technical difficulties.

4.2.1 Genericity assumptions and preliminary considerations

Let \mathcal{A}_{n}^{LC2}\subset\mathcal{A}_{n}^{\prime} be the set of polynomial foliations such that

  • g_{1}≔[\mathbf{M}_{1},\mathbf{M}_{2}]\in A_{1}, g_{2}≔[\mathbf{M}_{1}^{-1},\mathbf{M}_{2}]\in A_{1};

  • g_{3}≔[g_{2},g_{1}] is not the identity map2;

  • \mu_{1}\notin\mathbb{R};

  • the numbers 1, \mu_{1}, \mu_{1}^{-1}, \mu_{2}^{-1}, \mu_{1}^{-1}\mu_{2}^{-1}, \mu_{1}\mu_{2}^{-1} are all different.

\raisebox{10.0pt}{\hypertarget{id20}{}}\hyperlink{id19}{2}\raisebox{10.0pt}{\hypertarget{id20}{}}\hyperlink{id19}{2}footnotetext: Lemma 3 implies that for a generic \mathcal{F} we have g_{3}\in A_{3}, but our construction works for a slightly broader set of foliations.

Due to Lemma 3 and the fact that \sum\lambda_{i}=1 is the only relation on \lambda_{i}, the complement \mathcal{A}_{n}\smallsetminus\mathcal{A}_{n}^{LC2} is a real analytic subset of \mathcal{A}_{n}.

Consider a foliation \mathcal{F} from this set. Put g_{4}≔[g_{3},g_{2}]. Due to Lemma 2, g_{4}\neq\operatorname{id}.

Let \zeta be a Fatou chart for g_{1} in the negative sector.

Convention
In this Section, tilde above means that a map is written in the chart \zeta. In particular, {\tilde{g}}_{1}(\zeta)=\zeta+1.

Lemma 2 and Item b) of Lemma 1 imply

\displaystyle g_{1}(z) \displaystyle=z+\frac{g_{1}^{\prime\prime}(0)}{2}z^{2}+o(z^{2}) \displaystyle{\tilde{g}}_{1}(\zeta) \displaystyle=\zeta+1+o(1)
\displaystyle g_{2}(z) \displaystyle=z+\frac{g_{2}^{\prime\prime}(0)}{2}z^{2}+o(z^{2}) \displaystyle{\tilde{g}}_{2}(\zeta) \displaystyle=\zeta+\frac{g_{2}^{\prime\prime}(0)}{g_{1}^{\prime\prime}(0)}+o(1)
\displaystyle g_{3}(z) \displaystyle=z+az^{p+1}+o(z^{p+1}) \displaystyle{\tilde{g}}_{3}(\zeta) \displaystyle=\zeta-\frac{(-2)^{p}a}{g_{1}^{\prime\prime}(0)^{p}}\zeta^{1-p}+o% (\zeta^{1-p})
\displaystyle g_{4}(z) \displaystyle=z+\frac{a(p-1)g_{2}^{\prime\prime}(0)}{2}z^{p+2}+o(z^{p+2}) \displaystyle{\tilde{g}}_{4}(\zeta) \displaystyle=\zeta+\frac{(-2)^{p}a(p-1)g_{2}^{\prime\prime}(0)}{g_{1}^{\prime% \prime}(0)^{p+1}}\zeta^{-p}+o(\zeta^{-p}),

where a\in\mathbb{C}\smallsetminus\set{0}. Put {\tilde{a}}=-\frac{(-2)^{p}a}{g_{1}^{\prime\prime}(0)^{p}}, {\tilde{b}}=\frac{(-2)^{p}a(p-1)g_{2}^{\prime\prime}(0)}{g_{1}^{\prime\prime}(% 0)^{p+1}}. Due to (4), g_{2}^{\prime\prime}(0)=-\mu_{1}g_{1}^{\prime\prime}(0), hence

\displaystyle{\tilde{g}}_{1}(\zeta) \displaystyle=\zeta+1+o(1)
\displaystyle{\tilde{g}}_{2}(\zeta) \displaystyle=\zeta-\mu_{1}+o(1)
\displaystyle{\tilde{g}}_{3}(\zeta) \displaystyle=\zeta+\frac{{\tilde{a}}}{\zeta^{p-1}}+o\left(\frac{1}{\zeta^{p-1% }}\right)
\displaystyle{\tilde{g}}_{4}(\zeta) \displaystyle=\zeta+\frac{{\tilde{b}}}{\zeta^{p}}+o\left(\frac{1}{\zeta^{p}}% \right).

Since \frac{{\tilde{b}}}{{\tilde{a}}}=-\frac{(p-1)g_{2}^{\prime\prime}(0)}{g_{1}^{% \prime\prime}(0)}=(p-1)\mu_{1}\notin\mathbb{R}, each number T\in\mathbb{C} can be represented as

T=\xi(T){\tilde{a}}+\eta(T){\tilde{b}}. (8)

We will construct f_{j} as compositions of the maps g_{1},g_{2},g_{3},g_{4}.

4.2.2 Construction of f_{j}

Let \Delta^{+} be a small disc that we shall choose later. Now we just say that {\tilde{g}}_{2}^{\pm}\in\set{{\tilde{g}}_{2},{\tilde{g}}_{2}^{-1}} contracts in \Delta^{+},

\max_{\zeta\in\Delta^{+}}|({\tilde{g}}_{2}^{\pm})^{\prime}(\zeta)|=q<1. (9)

Since {\tilde{g}}_{2}(\zeta)-\zeta\rightarrow-\mu_{1} as \zeta\rightarrow\infty, we may and will assume that |{\tilde{g}}_{2}^{\pm}(\zeta)-\zeta|<|\mu_{1}|+1 for \zeta\in\Delta^{+}.

As in the previous case, take a disc \Delta^{-}\subset\Delta^{+} and a tuple of vectors T_{j}\in\mathbb{C} such that

\Delta^{-}\subset\bigcup_{j}({\tilde{g}}_{2}^{\pm}(\Delta^{-})+T_{j})\subset% \bigcup_{j}({\tilde{g}}_{2}^{\pm}(\Delta^{+})+T_{j})\subset\Delta^{+}. (10)

It is easy to see that the right inclusion implies |T_{j}+\mu_{1}|<2 or |T_{j}-\mu_{1}|<2, hence |T_{j}|<|\mu_{1}|+2. Put \xi_{j}=\xi(T_{j}), \eta_{j}=\eta(T_{j}), see (8).

Similarly to the previous case, we will choose the compositions {\tilde{f}}_{j} so that they will approximate the maps \zeta\mapsto{\tilde{g}}_{2}^{\pm}(\zeta)+T_{j} in C^{1}(\Delta^{+}). It turns out that we can use the compositions {\tilde{f}}_{j}={\tilde{F}}_{j}\circ{\tilde{g}}_{2}^{\pm}, where

{\tilde{F}}_{j}≔{\tilde{g}}_{1}^{-N}\circ{\tilde{g}}_{3}^{k_{j}}\circ{\tilde{g% }}_{4}^{l_{j}}\circ{\tilde{g}}_{1}^{N},\quad k_{j}=[N^{p-1}\xi_{j}],\quad l_{j% }=[N^{p}\eta_{j}]. (11)

Here N is a large number that we will choose later.

Let us prove that {\tilde{F}}_{j} approximate the translations \zeta\mapsto\zeta+T_{j} in C^{1}({\tilde{g}}_{2}^{\pm}(\Delta^{+})).

Lemma 7.

For N large enough, each head {\tilde{F}}_{j}^{(h)} such that \left(F_{j}^{(h)}\right)^{\prime}(0)=1 is close to a translation \zeta\mapsto\zeta+T in C^{1}({\tilde{g}}_{2}^{\pm}(\Delta^{+})). Moreover, \operatorname{Re}T>-|\operatorname{Re}\mu_{1}|-2 and |\operatorname{Im}T| is bounded by a number that does not depend on \Delta^{+}.

In particular,

  • {\tilde{g}}_{1}^{n} is the translation by n;

  • {\tilde{g}}_{4}^{n}\circ{\tilde{g}}_{1}^{N}, |n|\leqslant|k_{j}| is close to the translation by N+\frac{n{\tilde{b}}}{N^{p}};

  • {\tilde{g}}_{3}^{n}\circ{\tilde{g}}_{4}^{l_{j}}\circ{\tilde{g}}_{1}^{N}, |n|\leqslant|k_{j}| is close to the translation by N+\frac{n{\tilde{a}}}{N^{p-1}}+{\tilde{b}}\eta_{j};

  • {\tilde{g}}_{1}^{-n}\circ{\tilde{g}}_{3}^{k_{j}}\circ{\tilde{g}}_{4}^{l_{j}}% \circ{\tilde{g}}_{1}^{N}, 0\leqslant n\leqslant N, is close to the translation by T_{j}+N-n.

Proof.

We shall prove this lemma only for \xi_{j}>0 and \eta_{j}>0. For other cases, it is enough to replace {\tilde{g}}_{3} and (or) {\tilde{g}}_{4} by its inverse map.

Let us prove the assertions of the lemma for {\tilde{F}}_{j}^{(h)}={\tilde{g}}_{4}^{l_{j}}\circ{\tilde{g}}_{1}^{N}. Recall that {\tilde{g}}_{4}(\zeta)-\zeta={\tilde{b}}\zeta^{-p}+o(\zeta^{-p}), hence

{\tilde{F}}_{j}^{(h)}(\zeta)={\tilde{g}}_{4}^{l_{j}}(\zeta+N)=\zeta+N+\frac{{% \tilde{b}}l_{j}}{(\zeta+N)^{p}}+o(1)=\zeta+N+\frac{{\tilde{b}}[N^{p}\eta_{j}]}% {N^{p}}+o(1)=\zeta+N+{\tilde{b}}\eta_{j}+o(1)

as N\rightarrow\infty, \zeta\in{\tilde{g}}_{2}^{\pm}(\Delta^{+}). Therefore, {\tilde{F}}_{j}^{(h)} is C^{0}-close to the translation by N+{\tilde{b}}\eta_{j}.

Let us prove that the derivative of {\tilde{F}}_{j}^{(h)} is close to one. Due to Item c) of Lemma 1, \log{\tilde{g}}_{4}^{\prime}(\zeta)=o({\tilde{g}}_{4}(\zeta)-\zeta)=o(\zeta^{-% p}) as \zeta\rightarrow\infty, hence

\log\left({\tilde{F}}_{j}^{(h)}\right)^{\prime}(\zeta)=\sum_{k=0}^{l_{j}-1}% \log{\tilde{g}}_{4}^{\prime}({\tilde{g}}_{4}^{k}(\zeta+N))\leqslant N^{p}\eta_% {j}\log{\tilde{g}}_{4}^{\prime}(N+O(1))=o(1)

as N\rightarrow\infty, \zeta\in\Delta^{+}. Thus

\left({\tilde{F}}_{j}^{(h)}\right)^{\prime}(\zeta)=1+o(1).

Finally, in this case {\tilde{F}}_{j}^{(h)} is C^{1}-close to the translation by N+{\tilde{b}}\eta_{j}.

All particular cases listed in the statement of the lemma can be proved in the same way. Also, the estimate |T_{j}|<|\mu_{1}|+2 yields a uniform estimate on the imaginary parts of the translation vectors.

Consider a head {\tilde{F}}_{j}^{(h)}, \left(F_{j}^{(h)}\right)^{\prime}(0)=1, not listed explicitly in the statement of the lemma. Since g_{1} and g_{2} have no heads g with g^{\prime}(0)=1, {\tilde{F}}_{j}^{(h)} differs from a head of type b) or c) by a composition with a head {\tilde{g}} either of {\tilde{g}}_{3}^{\pm 1}, or of {\tilde{g}}_{4}^{\pm 1} such that g^{\prime}(0)=1. Since {\tilde{g}} is applied to points \zeta with \operatorname{Re}\zeta=N+O(1), it can be made arbitrarily C^{1}-close to its “translational part” \zeta\mapsto\zeta+T, T=\lim_{\zeta\rightarrow\infty}{\tilde{g}}(\zeta)-\zeta. Thus {\tilde{F}}_{j}^{(h)} is close to a translation \zeta\mapsto\zeta+T^{\prime} with bounded \operatorname{Im}T^{\prime} as well. ∎

4.2.3 Choice of \Delta^{+}

The construction relies on the following simple observation.

Lemma 8.

For a collection of hyperbolic maps F_{j}:(\mathbb{C},0)\rightarrow(\mathbb{C},0), F_{j}^{\prime}(0)\neq 1, there exists an arbitrarily thick strip

U=\Set{\zeta}{\operatorname{Re}\zeta>R,n_{-}<\operatorname{Im}\zeta<n_{+}},% \quad n_{+}-n_{-}>C, (12)

such that U does not overlap its images under {\tilde{F}}_{j}.

Proof.

Recall that {\tilde{F}}_{j}(\zeta)=k_{j}\zeta+b_{j}+o(1), see Lemma 1. For a map F_{j} with k_{j}\in\mathbb{R}, the affine term \zeta\mapsto k_{j}\zeta+b_{j} of {\tilde{F}} has invariant horizontal line \operatorname{Im}\zeta=y_{j}≔\frac{\operatorname{Im}b_{j}}{1-k_{j}}, and for |\operatorname{Im}\zeta-y_{j}|>\frac{C}{|k-1|} we have |\operatorname{Im}\zeta-\operatorname{Im}(k\zeta+b)|>C. Consider a strip U such that |\operatorname{Im}\zeta-y_{j}|>\frac{C}{|k_{j}-1|} whenever k_{j}\in\mathbb{R}. Clearly, for R large enough, all maps {\tilde{F}}_{j} will be close enough to their respective affine terms, hence {\tilde{F}}_{j}(U)\cap U=\varnothing. Finally, we enlarge R so that the assertion is satisfied for the maps {\tilde{F}}_{j} with k_{j}\notin\mathbb{R}. ∎

Let C_{1} be the estimate on |\operatorname{Im}T| from Lemma 7; put C_{2}=\max(C_{1},|\operatorname{Re}\mu_{1}|+2). Fix a strip (12), C=2C_{2}+|\operatorname{Im}\mu_{1}|, such that

\left(f_{i}^{(h)}\right)^{\prime}(0)\neq\left(f_{j}^{(h)}\right)^{\prime}(0)% \quad\Rightarrow\quad\left(f_{i}^{(h)}\right)^{-1}\circ\left(f_{j}^{(h)}\right% )(U)\cap U=\varnothing. (13)

Recall that {\tilde{g}}_{2}(\zeta)=\zeta-\mu_{1}+o(1). Hence there exists a small disc \Delta\subset U such that the distance between \mathbb{C}\smallsetminus U and \Delta\cup{\tilde{g}}_{2}(\Delta) is greater than C_{2}, and |{\tilde{g}}_{2}(\zeta)-\zeta+\mu_{1}|<1 for \zeta\in\Delta. Shrinking \Delta if necessary, we may and will assume that \forall\zeta\in\Delta we have |{\tilde{g}}_{2}^{\prime}(\zeta)|\neq 1. If |{\tilde{g}}_{2}^{\prime}(\zeta)|<1 in \Delta, then we put \Delta^{+}=\Delta, g_{2}^{\pm}=g_{2}, otherwise we put \Delta^{+}={\tilde{g}}_{2}(\Delta) and g_{2}^{\pm}=g_{2}^{-1}. Then {\tilde{g}}_{2}^{\pm} contracts in \Delta^{+}, see (9).

Finally, since the distance between \mathbb{C}\smallsetminus U and \Delta\cup{\tilde{g}}_{2}(\Delta) is greater than C_{2}, Lemma 7 implies that for N large enough, for each head {\tilde{F}}_{j}^{(h)} of {\tilde{F}}_{j}

\left(F_{j}^{(h)}\right)^{\prime}(0)=1\quad\Rightarrow\quad{\tilde{F}}_{j}^{(h% )}(\Delta^{+})\subset U,\qquad\left|\left({\tilde{F}}_{j}^{(h)}\right)^{\prime% }(\zeta)\right|<\frac{1}{\sqrt{q}},\qquad\left|\left({\tilde{F}}_{j}^{(t)}% \right)^{\prime}(\zeta)\right|<\frac{1}{\sqrt{q}} (14)

for \zeta\in{\tilde{g}}_{2}^{\pm}(\Delta^{+}), where q is given by (9).

4.2.4 Proof of the assumptions

Let us prove that for N large enough, the compositions {\tilde{f}}_{j} satisfy the assumptions listed in the plan of the proof. For assumptions inclusion and covering, this immediately follows from Lemma 7 and the definition of T_{j}.

Let us prove that (13) and (14) imply the contraction property. Consider a composition of the form {\tilde{f}}_{i}^{(t)}\circ{\tilde{f}}_{j}^{(h)}. Recall that {\tilde{f}}_{i} and {\tilde{f}}_{j} are compositions of the commutators {\tilde{g}}_{1}^{\pm 1} and {\tilde{g}}_{2}^{\pm 1}. Therefore, we can rewrite {\tilde{f}}_{i}^{(h)}, {\tilde{f}}_{j}^{(h)} and the corresponding tails as

\displaystyle{\tilde{f}}_{i}^{(h)} \displaystyle=\left({\tilde{g}}_{k}^{\pm 1}\right)^{(h)}\circ{\tilde{f}}_{i}^{% (ph)} \displaystyle{\tilde{f}}_{i}^{(t)} \displaystyle={\tilde{f}}_{i}^{(pt)}\circ\left({\tilde{g}}_{k}^{\pm 1}\right)^% {(t)},
\displaystyle{\tilde{f}}_{j}^{(h)} \displaystyle=\left({\tilde{g}}_{l}^{\pm 1}\right)^{(h)}\circ{\tilde{f}}_{j}^{% (ph)}, \displaystyle{\tilde{f}}_{j}^{(t)} \displaystyle={\tilde{f}}_{j}^{(pt)}\circ\left({\tilde{g}}_{k}^{\pm 1}\right)^% {(t)},

where

  • f_{i}^{(ph)}, f_{i}^{(pt)}, f_{j}^{(ph)}, f_{j}^{(pt)} are compositions of g_{1}^{\pm 1} and g_{2}^{\pm 1};

  • \set{k,l}\subset\set{1,2};

  • \left(g_{k}^{\pm 1}\right)^{(h)} and \left(g_{l}^{\pm 1}\right)^{(h)} may be empty but may not coincide with g_{k}^{\pm 1} or g_{l}^{\pm 1}.

If the maps \left(g_{k}^{\pm 1}\right)^{(h)} and \left(g_{l}^{\pm 1}\right)^{(h)} have different multipliers, then (13) and (14) imply that (f_{j}^{(h)})^{-1}\circ f_{i}^{(h)}(\Delta^{+})\cap\Delta^{+}=\varnothing.

Next, suppose that \left(g_{k}^{\pm 1}\right)^{(h)} and \left(g_{l}^{\pm 1}\right)^{(h)} have equal multipliers. It is easy to check that our assumption on \mu_{1}, \mu_{2} implies that in this case \left(g_{k}^{\pm 1}\right)^{(t)}\circ\left(g_{l}^{\pm 1}\right)^{(h)} is one of the maps \operatorname{id}, g_{k}^{\pm 1}, g_{l}^{\pm 1}. In the first case, we just eliminate the middle part from

f_{i}^{(t)}\circ f_{j}^{(h)}={\tilde{f}}_{i}^{(pt)}\circ\left[\left({\tilde{g}% }_{k}^{\pm 1}\right)^{(t)}\circ\left({\tilde{g}}_{l}^{\pm 1}\right)^{(h)}% \right]\circ{\tilde{f}}_{j}^{(ph)},

and in the two latter cases we can regard the middle part either as a part of f_{i}^{(t)}, or as a part of f_{j}^{(h)}. Hence we can assume that both f_{i}^{(t)} and f_{j}^{(h)} are parabolic maps.

Finally, due to (14), for parabolic f_{i}^{(t)} and f_{j}^{(h)} we have

\left|\left(f_{i}^{(t)}\circ f_{j}^{(h)}\right)^{\prime}(\zeta)\right|<q\times% \frac{1}{\sqrt{q}}\times\frac{1}{\sqrt{q}}=1.

Hence, the maps f_{j} satisfy the contraction requirement.

5 Construction of limit cycles

Consider a polynomial foliation \mathcal{F}\in\mathcal{A}_{n}^{\prime}. Suppose that there exist domains \Delta^{-}\subset\Delta^{+}, a chart \zeta and a tuple of monodromy maps f_{j} that satisfy the assumptions listed in the plan of the proof. In this section we shall show that such foliation satisfies the assertions of Main Theorem. The proof is based on the following simple observation.

Lemma 9.

Suppose that a collection of limit cycles c_{j} satisfies the following:

  • all cycles c_{j} are simple, i.e., have no self-intersections;

  • their multipliers \mu(c_{j}) satisfy 0<|\mu(c_{j})|<|\mu(c_{1})\cdots\mu(c_{j-1})|;

  • c_{i}\cap c_{j}=\varnothing for i\neq j.

Then these cycles are homologically independent.

Proof.

Since all these cycles are simple and do not intersect each other, a possible dependency has the form \pm[c_{i_{1}}]\pm[c_{i_{2}}]\pm\ldots\pm[c_{i_{s}}]=0, i_{1}<i_{2}<\ldots<i_{s}. However such dependence implies the equality on multipliers, \mu(c_{i_{1}})^{\pm 1}\mu(c_{i_{2}})^{\pm 1}\ldots\mu(c_{i_{s-1}})^{\pm 1}=\mu% (c_{i_{s}}), which is impossible due to the inequality

|\mu(c_{i_{s}})|<|\mu(c_{1})\ldots\mu(c_{i_{s}-1})|\leqslant|\mu(c_{i_{1}})% \ldots\mu(c_{i_{s-1})}|\leqslant|\mu(c_{i_{1}})^{\pm 1}\mu(c_{i_{2}})^{\pm 1}% \ldots\mu(c_{i_{s-1}})^{\pm 1}|.

Remark.

In earlier papers [Il78, SRO98] the authors used similar arguments, but they estimated \int_{c_{j}}x\,dy-y\,dx instead of multipliers. This led to much more complicated computations.

As we mentioned above, Lemma 5 enables us to construct cycles with arbitrarily small multipliers, but these cycles may be neither simple, nor disjoint. The following two lemmas fill these gaps.

Lemma 10.

Let D\subset\mathbb{R}^{n} be a closed disc, g_{1},g_{2}:D\rightarrow D be two injective continuous maps such that g_{1}(D)\cap g_{2}(D)=\varnothing, \Sigma\subset D be a finite subset. Then for m large enough there exists a periodic orbit

p_{0},p_{1},\ldots,p_{m}=p_{0},p_{i+1}\in\set{g_{1}(p_{i}),g_{2}(p_{i})} (15)

that never meets \Sigma.

Lemma 11.

Given

  • an open subset U\Subset\Delta^{-};

  • two maps g_{1}=f_{i_{1}}\circ\ldots\circ f_{i_{s}}, g_{2}=f_{j_{1}}\circ\ldots\circ f_{j_{r}}, g_{i}:\Delta^{+}\rightarrow U with disjoint images;

  • a finite set \Sigma\subset S;

  • a positive number \varepsilon,

there exists a finite set \Sigma^{\prime}\subset\Delta^{+} such that the following holds. Suppose that a periodic orbit (15) never visits \Sigma^{\prime}. Since p_{0} is a fixed point of some monodromy map, it corresponds to a cycle c. Let c^{\prime} be its simple subcycle. Then c^{\prime} visits U, never visits \Sigma, and the modulus of its multiplier is less than \varepsilon.

Let us deduce Main Theorem from these two lemmas.

Proof of Main Theorem.

Fix a sequence of points x_{k} in the interior of \Delta^{-} dense in \Delta^{-}. Let U_{k} be the intersection of \Delta^{-} with the (1/k)-neighborhood of x_{k}. Now we construct the sequence c_{j} by induction. Suppose that simple homologically independent cycles c_{1},c_{2},\ldots,c_{k-1} are already constructed and have multipliers \mu(c_{j}), |\mu(c_{j})|<1. Put \Sigma=\bigcup_{j}c_{j}\cap S, \varepsilon=|\mu(c_{1})\mu(c_{2})\cdots\mu(c_{k-1})|.

Take two disjoint domains V_{1},V_{2}\subset U_{k}. Due to Lemma 5, there exist two contracting compositions g_{1}:\Delta^{+}\rightarrow V_{1} and g_{2}:\Delta^{+}\rightarrow V_{2}. According to the previous two lemmas, there exists a simple cycle c_{k} with multiplier less than \varepsilon that intersects U_{k} but does not visit \Sigma. Note that this cycle, as well as all previous ones, projects to the curve of the form \gamma_{l_{1}}\gamma_{l_{2}}\ldots\gamma_{l_{r}} on the infinite line, and \gamma_{i}\cap\gamma_{j}=\set{O} for i\neq j. Thus if c_{i}\cap c_{k}\neq\varnothing for some i<k, then c_{k}\cap c_{i}\cap S\neq\varnothing, hence c_{k}\cap\Sigma\neq\varnothing which contradicts the choice of c_{k}. Due to Lemma 9, the cycles c_{1},\ldots,c_{k} are homologically independent. ∎

Now let us prove the lemmas formulated above.

Proof of Lemma 10.

Fix a large number m. Due to Brouwer Theorem, for each word w=w_{1}\ldots w_{m}, w_{i}\in\set{1,2}, the corresponding map

g_{w}=g_{w_{1}}\circ\ldots\circ g_{w_{m}}:D\rightarrow D

has a fixed point. Our goal is to find a word w such that the corresponding periodic orbit will never visit \Sigma. Since g_{1}(D)\cap g_{2}(D)=\varnothing, the images of 2^{m} maps g_{w}, |w|=m, are pairwise disjoint. Hence, given a point p\in\Sigma, there is at most one word w such that g_{w}(p)=p. Their cyclic shifts are the only words w such that the corresponding periodic orbit visits \Sigma, thus there are at most |\Sigma|\cdot m of them. Clearly, for m large enough we have |\Sigma|\cdot m<2^{m}, hence there exists a periodic orbit of length m that never visits \Sigma. ∎

Proof of Lemma 11.

Consider a composition g_{w}=g_{w_{1}}\circ\ldots\circ g_{w_{s}}=\mathbf{M}_{i_{1}}\circ\ldots\circ% \mathbf{M}_{i_{k}}, the corresponding periodic orbit (15) and the corresponding limit cycle c. If (15) does not visit the finite set \Sigma_{1}=\bigcup_{g_{i}^{(h)}}\left(g_{i}^{(h)}\right)^{-1}(\Sigma), then c\cap\Sigma=\varnothing.

Subcycles of c correspond to representations g_{w}=g^{(t)}\circ g^{(m)}\circ g^{(h)} with non-empty g^{(m)} such that g^{(h)}(p_{0}) is a fixed point of g^{(m)}.

Let us prove that sufficiently long compositions g^{(m)} correspond to cycles c^{\prime} that satisfy the assertions of the lemma, and fixed points of “short” compositions can be avoided by avoiding a finite set \Sigma_{2}.

If g^{(m)} is not a subcomposition of one of f_{j}, then it can be represented in the form

g^{(m)}=f_{s_{1}}^{(h)}\circ f_{s_{2}}\circ\ldots\circ f_{s_{k-1}}\circ f_{s_{% k}}^{(t)}. (16)

Recall that f_{s_{k}}^{(t)}\circ f_{s_{1}}^{(h)} contracts in the chart \zeta due to contraction property, thus (f_{s_{1}}^{(h)})^{-1}\circ g^{(m)}\circ f_{s_{1}}^{(h)} contracts, and we have

\mu(c^{\prime})\leqslant\left(\max_{\zeta\in\Delta^{+}}f_{i}^{\prime}(\zeta)% \right)^{k-2}. (17)

Let \operatorname{len}(\cdot) be the length of a composition of \mathbf{M}_{j}. For sufficiently large L, \operatorname{len}(g^{(m)})\geqslant L implies that the subcycle c^{\prime} corresponding to g^{(m)} satisfies the assertions of the lemma. Indeed, the multiplier of c^{\prime} can be made arbitrarily small due to (17); for any L>\max(\operatorname{len}g_{1},\operatorname{len}g_{2}), the corresponding c^{\prime} visits U because it contains a point of the form g_{w_{i+1}}\circ\ldots\circ g_{w_{s}}(p_{0})\in U.

Now, it is sufficient to avoid fixed points of “short” compositions g^{(m)}, \operatorname{len}(g^{(m)})<L. Let us prove that none of the compositions g^{(m)} are identical in g^{(h)}(\Delta^{+}). Suppose the contrary. We have eliminated all identical subcompositions from f_{j}, so g^{(m)} cannot be a subcomposition of some f_{j}. Thus g^{(m)} has the form (16), and (17) yields that g^{(m)} contracts. Hence g^{(m)}\neq\operatorname{id}.

Therefore, each composition g^{(m)} has only finitely many fixed points in g_{w_{j}}^{(h)}(\Delta^{+}), where g_{w_{j}}^{(h)}(\Delta^{+}) is defined by g^{(h)}=g_{w_{j}}^{(h)}\circ g_{w_{j+1}}\circ\ldots\circ g_{w_{s}}. In order to guarantee that a subcycle c^{\prime} corresponds to a long composition g^{(m)}, \operatorname{len}g^{(m)}>L, it is sufficient to require that the periodic orbit (15) avoids the finite set

\Sigma_{2}=\set{(g_{w_{j}}^{(h)})^{-1}\operatorname{Fix}g^{(m)}}{\operatorname% {len}(g^{(m)})<L}.

The required exceptional set is \Sigma^{\prime}=\Sigma_{1}\cup\Sigma_{2}. ∎

6 Acknowledgements

We proved these results and wrote the first version of this article during our 5-months visit to Mexico (Mexico City, then Cuernavaca). We are very grateful to UNAM (Mexico) and HSE (Moscow) for supporting this visit. Our deep thanks to Laura Ortiz Bobadilla, Ernesto Rosales-González and Alberto Verjovsky, for invitation to Mexico and for fruitful discussions.

We are thankful to Arsenij Shcherbakov for useful discussions about technical details of [SRO98]. We are also grateful to Yulij Ilyashenko for permanent encouragement, and to Victor Kleptsyn for interesting discussions.

References

  • [BLL97] M. Belliart, I. Liousse, F. Loray Sur l’existence de points fixes attractifs pour les sous-groupes de Aut(\mathbb{C},0) // C. r. Acad. sci. Paris. Sér. 1. 1997. V. 324, N 4. P. 443–446.
  • [H81] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30:5 (1981), 713–747.
  • [Il78] Yu. S. Ilyashenko, Topology of phase portraits of analytic differential equations on a complex projective plane, Trudy Sem. Petrovsk., Vol. 4 (1978), pp. 83–136
  • [IYa07] Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, Vol. 86, AMS, 2007
  • [KhV62] M. G. Khudaĭ-Verenov, A property of the solutions of a differential equation (in Russian), Math. USSR Sb., Vol. 56(98) (1962), pp. 301–308
  • [N94] I. Nakai Separatrices for non-solvable dynamics on (\mathbb{C},0) // Ann. Inst. Fourier. 1994. V. 44, N 2. P. 569–599.
  • [Shch06] A. Shcherbakov, Dynamics of local groups of conformal mappings and generic properties of differential equations on \mathbb{C}^{2}, Proceedings of the Steklov Institute of Mathematics, Vol. 254, pp. 103–120
  • [Shch84] A. Shcherbakov, Topological and analytical conjugacy of non-commutative groups of germs of conformal mappings, Trudy Sem. Petrovsk., Vol. 10 (1984), pp. 170–196
  • [SRO98] A. Shcherbakov, E. Rosales-González, L. Ortiz-Bobadilla, Countable set of limit cycles for the equation dw/dz=P_{n}(z,w)/Q_{n}(z,w), J. Dynam. Control Systems, Vol. 4 (1998), No. 4, pp. 539–581
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