Periodic complex map germs and foliations

In this paper we study topological and analytical conditions on the orbits of a germ of diffeomorphism in the complex plane in order to obtain periodicity. In particular, we give a simple proof of a finiteness criteria for groups of analytic diffeomorphisms, stated in Brochero Martínez 2003. As an application, we derive some consequences about the integrability of complex vector fields in dimension three in a neighborhood of a singular point.

Key words
Complex germs of diffeomorphims; singular holomorphic foliations; integrability of vector fields; closed orbits

Authorship

LEONARDO M. CÂMARA

Universidade Federal do Espírito Santo, Departamento de Matemática - CCE, Av. Fernando Ferrari, 514, 29075-910 Vitória, ES, BrazilUniversidade Federal do Espírito SantoBrazilVitória, ES, BrazilUniversidade Federal do Espírito Santo, Departamento de Matemática - CCE, Av. Fernando Ferrari, 514, 29075-910 Vitória, ES, Brazil

BRUNO A. SCÁRDUA

Universidade Federal do Rio de Janeiro, Instituto de Matemática, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, BrazilUniversidade Federal do Rio de JaneiroBrazilRio de Janeiro, RJ, BrazilUniversidade Federal do Rio de Janeiro, Instituto de Matemática, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil

Correspondence to: Leonardo Meireles Câmara E-mail: leonardo.camara@ufes.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (3)
Formulas (23)

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Figure 1
Blowing-up infinitely many separatrices.

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Figure 2
A dicritical component of ℱ.

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Figure 3
The liftings of γ along the leaves of 𝔉X starting at points of L.

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(1)

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(2)

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(3)

Figure 1 Blowing-up infinitely many separatrices.

Figure 2 A dicritical component of ℱ.

Figure 3 The liftings of γ along the leaves of 𝔉X starting at points of L.

𝒪_{V}^{+} (G, x) = {G^{\circ (n)} (x) : G^{\circ (j)} (x) \in V, j = 0, \dots, n}

\tilde{g} (t, x) = (\frac{g_{2} (x, t x)}{g_{1} (x, t x)}, g_{1} (x, t x)) = (\frac{α_{2} + β_{2} t}{α_{1} + β_{1} t} + x (\dots), g_{1} (x, t x)) .

(λ^{m} s^{m} + \dots, μ λ^{n} s^{n} + \dots) = \tilde{g} (s^{m}, μ s^{n} + \dots) = (s^{m} + \dots, α μ s^{n} + \dots),

\tilde{G} (t, x)

= (\frac{g_{2} (x, t x)}{g_{1} (x, t x)}, g_{1} (x, t x)) = (\frac{t}{1 + t^{2} x}, x + t^{2} x^{2})

= (t (1 - t^{2} x^{2} + t^{4} x^{4} - t^{6} x^{6} + \dots), x (1 + t^{2} x))

= (t - t^{3} x^{2} + t^{5} x^{4} - t^{7} x^{6} + \dots, x + t^{2} x^{2})

X (x) = - m_{1} [x_{1} (1 + a_{1} (x)) + x_{2} b_{1} (x)] \frac{\partial}{\partial x_{1}} - m_{2} x_{2} (1 + a_{2} (x)) \frac{\partial}{\partial x_{2}} + x_{3} \frac{\partial}{\partial x_{3}}

\frac{\partial}{\partial t} {\bar{Γ}}_{(x_{1}, x_{2})} (t) = α X (Γ_{1} (t, x_{1}, x_{2}), Γ_{2} (t, x_{1}, x_{2}), γ (t)) .

\frac{\partial}{\partial t} Γ_{1}

(1)

= - 2 m_{1} π 𝐢 [Γ_{1} \cdot (1 + a_{1} (Γ_{1}, Γ_{2}, γ)) + Γ_{2} \cdot b_{1} (Γ_{1}, Γ_{2}, γ)],

\frac{\partial}{\partial t} Γ_{2}

(2)

= - 2 m_{2} π 𝐢 Γ_{2} \cdot (1 + a_{2} (Γ_{1}, Γ_{2}, γ)) .

\frac{\partial}{\partial t} Γ_{2} = - 6 π 𝐢 Γ_{2} .

\frac{\partial}{\partial t} Γ_{1} = - 2 π 𝐢 [Γ_{1} + e^{- 6 π 𝐢 t} x_{2}^{2} b (γ (t))] = - 2 π 𝐢 (Γ_{1} + e^{- 6 π 𝐢 t} b (e^{2 π 𝐢 t}) \cdot x_{2}^{2}) .

α^{'} (t) = - 2 π 𝐢 \cdot α (t) - 2 π 𝐢 e^{- 6 π 𝐢 t} b (e^{2 π 𝐢 t}), α (0) = 0

α (t) = - 2 π 𝐢 e^{- 2 π 𝐢 t} \int_{0}^{t} e^{2 π 𝐢 s} e^{- 6 π 𝐢 s} b (e^{2 π 𝐢 s}) 𝑑 s = - e^{- 2 π 𝐢 t} \int_{0}^{t} e^{- 6 π 𝐢 s} b (e^{2 π 𝐢 s}) 2 π i e^{2 π 𝐢 s} 𝑑 s

π^{*} X (t, x, z)

= - (x - \frac{1}{2 π 𝐢} t^{2} x^{2} z^{2}) \frac{\partial}{\partial x} + \frac{1}{x} (- 3 t x - t (- x + t^{2} x^{2} z^{2})) \frac{\partial}{\partial t} + z \frac{\partial}{\partial z}

= - x (1 - \frac{1}{2 π 𝐢} t^{2} x z^{2}) \frac{\partial}{\partial x} - t (2 + t^{2} x z^{2}) \frac{\partial}{\partial t} + z \frac{\partial}{\partial z}

X (x, y, z) = X_{n, m, - k} := n x \frac{\partial}{\partial x} + m y \frac{\partial}{\partial y} - k \frac{\partial}{\partial z}

ϕ (t) = (x (t), y (t), z (t)) = (a e^{n t}, b y e^{m t}, c e^{- k t}), t \in ℂ .

(3)

X = m x (1 + a (x, y, z)) \frac{\partial}{\partial x} + n y (1 + b (x, y, z)) \frac{\partial}{\partial x} - k z (1 + c (x, y, z) \frac{\partial}{\partial z}

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Periodic complex map germs and foliations

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[1] Correspondence to: Leonardo Meireles Câmara E-mail: leonardo.camara@ufes.br