2012/02/10 by Manuel D. Contreras, Santiago Diaz-Madrigal, Contreras, Manuel D. +4 · 2 citations
Computer Science · Mathematics · #30C35 #30C80 #30D05 #34M15 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Differential Equations Analysis #Nonlinear Dynamics and Pattern Formation #math.CA #math.CV #math.DS #msc:30C35 #msc:30C80 #msc:30D05 #msc:34M15
paper · pdf · doi:10.48550/arxiv.1202.2334
openalex publication_date 2012/02/10 · arxiv created 2012/02/27 · arxiv updated 2012/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Among diversity of frameworks and constructions introduced in Loewner Theory by different authors, one can distinguish two closely related but still different ways of reasoning, which colloquially may be described as "increasing" and "decreasing". In this paper we review in short the main types of (deterministic) Loewner evolution discussed in the literature and describe in detail the local duality between "increasing" and "decreasing" cases within the general unifying approach in Loewner Theory proposed recently in [Bracci et al. to appear in J Reine Angew Math; arXiv:0807.1594v1], [Bracci et al. in Math Ann 344:947-962, 2009; arXiv:0807.1715v1], [Contreras et al. in Revista Matemática Iberoamericana 26:975-1012, 2010; arXiv:0902.3116v1]. In particular, we extend several results of R.O.Bauer [in J Math Anal Appl 302:484-501, 2005; arXiv:math/0306130v1], which deal with the chordal Loewner evolution, to this general setting. Although the duality is given by a simple change of the parameter, not all the results for the "decreasing" case can be obtained by mere translating the corresponding results for the "increasing" case. In particular, as a byproduct of establishing local duality between evolution families and their "decreasing" counterparts we obtain a new characterization of generalized Loewner chains.