In this note, we will consider the question of local equivalence of analytic functions which fix the origin and are tangent to the identity, as well as the question of flows of analytic vector fields. All mappings and equivalences are considered in the non-archimedean context e.g. all norms can be considered p-adic norms. We show that any two mappings f and g which are formally equivalent are also analytically equivalent, and we show that analytic vector fields generate analytic flows. We consider the related questions of roots and centralizers for analytic mappings. In this setting, anything which can be done formally can also be done analytically.
Abstract. In this paper, we will consider (germs of) holomorphic mappings of the form (f (z), λ 1 w 1 (1 + g 1 (z)), . . . , λ n w n (1 + g n (z))), defined in a neighborhood of the origin in C n+1 . Most of our interest is in those mappings where f (z) = z + a m z m + · · · is a germ tangent to the identity and g i (0) = 0 for i = 1, . . . , n, and λ i ∈ C possess no resonances, for these are the so-called Poincaré-Dulac normal forms of the mappings (z + O(2), λ 1 w + O(2), . . . , λ n w + O(2)). We construct formal normal forms for these mappings and discuss a condition which tests for the convergence or divergence of the conjugating maps, giving specific examples.
We study the existence of solutions of the generalized Beltrami equation fz = µ(z)fz, µ(z) ∞ = 1, in a plane domain ∆, under general conditions that include previously known results.
Abstract. We consider germs of conformal mappings tangent to the identity at the origin in C. We construct a germ of a homeomorphism which is a C ∞ diffeomorphism except at the origin conjugating these holomorphic germs with the time-one map of the vector field V (z) = z m ∂ ∂z . We then show that, in the case m = 2, for a germ of a homeomorphism which is real-analytic in a punctured neighborhood of the origin, with real-analytic inverse, conjugating these germs with the time-one map of the vector field exists if and only if a germ of a biholomorphism exists.
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