# Relations of Formal Diffeomorphisms and the Center Problem

@article{Nakai2010RelationsOF, title={Relations of Formal Diffeomorphisms and the Center Problem}, author={Isao Nakai and Kana Yanai}, journal={Moscow Mathematical Journal}, year={2010}, volume={10}, pages={415-468} }

A word of germs of holomorphic diffeomorphisms of (C, 0) is a composite of some time-1 maps of formal vector fields fixing 0, in other words, a noncommutative integral of a piecewise constant time depending formal vector field. We calculate its formal-vector-field-valued logarithm applying the Campbell–Hausdorff type formula of the Lie integral due to Chacon and Fomenko to the time depending formal vector field. For words of two time 1-maps we define Cayley diagrams in the plane spanned by the…

## 8 Citations

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. The purpose of this note is to give a fresh insight into the problem of relations of two germs of holomorphic diﬀeomorhisms from the view point of the center problem of ordinary diﬀerential…

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A solution $$y(x)$$ of an Abel differential equation $$(1) \ y^{\prime }=p(x)y^2 + q(x) y^3$$ is called “closed” on $$[a,b]$$ if $$y(a)=y(b)$$. The equation $$(1)$$ is said to have a center on…

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We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated…

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### Ergodic Theory and Dynamical Systems

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The Abel differential equation y = p(x)y3 + q(x)y2 with polynomial coefficients p, q is said to have a center on [a, b] if all its solutions, with the initial value y(a) small enough, satisfy the…

### Algebraic geometry of the center-focus problem for Abel differential equations

- MathematicsErgodic Theory and Dynamical Systems
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The Abel differential equation $y^{\prime }=p(x)y^{3}+q(x)y^{2}$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small…

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