Basins of attraction in Loewner equations

2011/08/30 by Arosio, Leandro
#37F99 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 32H50 #Secondary 32H02

paper · doi:10.48550/arxiv.1108.6000

Abstract

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))≥ k(Dh(0,t)) for some r∈ R⁺, admits a solution given by univalent mappings (f_t: B^q→ 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.

Related

• Embedding Problems in Loewner Theory
• Loewner equations on complete hyperbolic domains
• Loewner chains in the unit disk
• Explicit multi-slit Loewner flows and their geometry
• Solving the Loewner PDE in complete hyperbolic starlike domains of\n C^N