Basins of attraction in Loewner equations
Copyright policy )Basins of attraction in Loewner equations
We prove that any Loewner PDE whose driving term h(z,t) vanishes at the origin, and satisfies the bunching condition r m(Dh(0,t))\geq k(Dh(0,t)) for some r\in R^+, admits a solution given by univalent mappings (f_t: B^q\to C^q). This is done by discretizing time and considering the abstract basin of attraction. If r<2, then the union of the images f_t(\B^q) of a such solution is biholomorphic to C^q.
9 pages, revised exposition, improved results, added references
Settore MAT/03 - GEOMETRIA, infinitesimal generator, Mathematics - Complex Variables, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Primary 32H50, Secondary 32H02, 37F99, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Dynamical Systems (math.DS), Evolution family, Herglotz vector field, Loewner differential equation, univalent mapping, Dynamical systems over complex numbers, FOS: Mathematics, Loewner chain of order \(\infty\), Mathematics - Dynamical Systems, Complex Variables (math.CV)
Settore MAT/03 - GEOMETRIA, infinitesimal generator, Mathematics - Complex Variables, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Primary 32H50, Secondary 32H02, 37F99, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Dynamical Systems (math.DS), Evolution family, Herglotz vector field, Loewner differential equation, univalent mapping, Dynamical systems over complex numbers, FOS: Mathematics, Loewner chain of order \(\infty\), Mathematics - Dynamical Systems, Complex Variables (math.CV)
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