Annales Academię Scientiarum Fennicę
Mathematica
Volumen 37, 2012, 563-570

BASINS OF ATTRACTION IN LOEWNER EQUATIONS

Leandro Arosio

Istituto Nazionale di Alta Matematica "Francesco Severi"
Cittą Universitaria, Piazzale Aldo Moro 5, 00185 Rome, Italy; arosio 'at' altamatematica.it

Abstract. Let q \geq 2. We prove that any Loewner PDE on the unit ball B^q whose driving term h(z,t) vanishes at the origin and satisfies the bunching condition lm(Dh(0,t)) \geq k(Dh(0,t)) for some l \in R⁺, admits a solution given by univalent mappings (f_t : B^q \to C^q)_{t \geq 0}. This is done by discretizing time and considering the abstract basin of attraction. If l < 2, then the range \cup_{t \geq 0} f_t(B^q) of any such solution is biholomorphic to C^q.

2010 Mathematics Subject Classification: Primary 32H50; Secondary 32H02, 37F99.

Key words: Loewner chains in several variables, Loewner equations, evolution families, abstract basins of attraction.

Reference to this article: L. Arosio: Basins of attraction in Loewner equations. Ann. Acad. Sci. Fenn. Math. 37 (2012), 563-570.

Full document as PDF file

doi:10.5186/aasfm.2012.3742

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