Relations of formal diffeomorphisms and the center problem

A word of germs of holomorphic diffeomorphisms of $(\mathbb C,0)$ 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 generating two vector fields in the Lie algebra of formal vector fields, and we show that some principal parts in the Taylor coefficients of the logarithm are given in terms of the higher moments of the Cayley diagrams. Solving the so-called center problem, the vanishing of the Lie integral, we show the various results on the existence and non-existence of relations of non-commuting formal diffeomorphisms in terms of the characteristic curves associated to the Cayley diagram.