Bounded Limit Cycles of Polynomial Foliations of $\mathbb {C}^{2}$

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.

Appendix A: Closing a Gap in (Shcherbakov et al. 1998)

Recall the construction of the cycles from Shcherbakov et al. (1998) in terms of the present article. Consider⁴ U, $g_i$, $\Delta ^{+}$, N and one of $f_j$ (say, $f=f_1$) from Section 4.2.

Since ${\tilde{f}}:\Delta ^{+}\rightarrow \Delta ^{+}$ contracts, it has a fixed point $\zeta _{0}\in \Delta ^{+}$. Let c be the corresponding limit cycle. The authors of Shcherbakov et al. (1998) state that c is a simple cycle, and prove that $\int _c x\,dy-y\,dx$ can be made arbitrarily large.

Unfortunately, there is a gap in the proof of simplicity of c. Namely, in §3.4, the authors say that the projection $\gamma :(\mathbb {R}/\mathbb {Z}, 0)\rightarrow (\mathbb {C}P^{1}{\smallsetminus } \left\{ a_{0}, \ldots , a_n\right\} , O)$ of c to the infinite line can be deformed so that all its self-intersections are located in an arc $\gamma ([1-\varepsilon , 1])$ that does not leave a small neighbourhood of the initial point O. This is not true, say, for $\gamma =[\gamma _{1},[\gamma _{1}, \gamma _{2}]]$. Therefore, one needs more arguments to prove that c has no self-intersections.

It seems that these cycles are simple, but the proof of this fact needs a tedious case-by-case analysis, and we did not manage to check all cases. In this Section, we shall prove that simple subcycles of the cycles constructed in Shcherbakov et al. (1998) are homologically independent.

Similarly to Lemma 5.1, it is enough to prove that there exists a simple cycle arbitrarily close to the infinite line with multiplier arbitrarily close to one. Indeed, in this case we just choose the next cycle $c_j$ so that

• 1. it is closer to the infinite line than all the previous cycles;

• 2. $\mu (c_j)$ is closer to one than all the numbers $\mu (c_{i_1})^{\pm 1}\ldots \mu (c_{i_s})^{\pm 1}$, $1\leqslant i_1<\cdots<i_s<j$.

Let us prove that for $\min \mathrm{Re}(\Delta ^{+})$ and N large enough, any simple subcycle $c'$ of c will satisfy the above assumptions. For the first assumption, it is enough to take large $\min \mathrm{Re}(\Delta ^{+})$.

As we noted in the proof of Lemma 5.3, a self-intersection of c corresponds to a splitting of ${\tilde{f}}$ into three compositions, ${\tilde{f}}={\tilde{f}}^{(t)}\circ {\tilde{f}}^{(m)}\circ {\tilde{f}}^{(h)}$ such that ${\tilde{f}}^{(h)}(\zeta _{0})$ is a fixed point of ${\tilde{f}}^{(m)}$. The self-intersection provides two subcycles, one is generated by $\left( {\tilde{f}}^{(t)}\circ {\tilde{f}}^{(h)}\right) (\zeta _{0})=\zeta _{0}$, and the other one is generated by ${\tilde{f}}^{(m)}(\zeta _{1})=\zeta _{1}$, where $\zeta _{1}={\tilde{f}}^{(h)}(\zeta _{0})$.

Note that $\left( {\tilde{f}}^{(m)}\circ {\tilde{f}}^{(h)}\right) (\zeta _{0})$ belongs to ${\tilde{f}}^{(h)}(\Delta ^{+})$, hence

$$\begin{aligned} \zeta _{0}\in \left( \left( {\tilde{f}}^{(h)}\right) ^{-1}\circ \left( {\tilde{f}}^{(m)}\circ {\tilde{f}}^{(h)}\right) \right) (\Delta ^{+})\cap \Delta ^{+}\ne \varnothing . \end{aligned}$$

As we noted in Sect. 4.2.4, (10) and (11) now imply that

$$\begin{aligned} \left( f^{(m)}\circ f^{(h)}\right) '(0)=\left( f^{(h)}\right) '(0), \end{aligned}$$

thus $f^{(m)}$ is tangent to identity. Proceeding in the same way as in Sect. 4.2.4, we can show that ${\tilde{f}}$ possesses a representation ${\tilde{f}}={\tilde{f}}^{(t)}\circ {\tilde{f}}^{(m)}\circ {\tilde{f}}^{(h)}$ with ${\tilde{f}}^{(m)}\left( {\tilde{f}}^{(h)}(\zeta _{0})\right) ={\tilde{f}}^{(h)}(\zeta _{0})$, $f^{(h)}$ tangent to identity, and ${\tilde{f}}^{(m)}$ conjugated to the original one.

The multipliers of the cycles corresponding to the self-intersection point ${\tilde{f}}^{(h)}(\zeta _{0})$ are

$$\begin{aligned} \left( {\tilde{f}}^{(m)}\right) '\left( {\tilde{f}}^{(h)}(\zeta _{0})\right) \text { and }\left( {\tilde{f}}^{(h)}\circ {\tilde{f}}^{(t)}\right) '\left( \left( f^{(t)}\right) ^{-1}(\zeta _{0})\right) . \end{aligned}$$

Due to Lemma 4.2, both of these numbers tend to $\left\{ 1\right\} \cup {\tilde{g}}_{2}'(\Delta ^{+})$ as $N\rightarrow \infty$. Recall that ${\tilde{g}}_{2}'(\zeta )\rightarrow 1$ as $\zeta \rightarrow \infty$, hence the multipliers of all simple subcycles of c tend to one as $\min \mathrm{Re}(\Delta ^{+})$ and N tend to infinity.

Finally, we constructed a simple cycle arbitrarily close to the infinite line with multiplier arbitrarily close to 1, hence $\mathcal {F}$ possesses infinitely many homologically independent limit cycles.