The compactification of the two dimensional monomial map

• Autor(a) principal: Silva, Samanta Santos Avelino
• Data de Publicação: 2023
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-09102023-230758/
• topic: Blow-ups; Blow-ups; Compactificação; Compactification; Continued fractions; Frações contínuas; Mapa monomial; Monomial map

Given a matrix = ( & \\\\ & ) in 2(), we can define its associated monomial map _ ^2 ^2 as follows: _ (,) = (^ ^,^ ^ ) . In the open set (^)^2, _ is biholomorphic and its dynamics are well known (Bonnot et al., 2018). However, as discussed by Favre, 2003, the dynamics can also be extended to ^2 through toric geometry compactification. This method, while precise, can be somewhat technical. Our goal is to provide a simpler, alternative approach to the compactification problem that achieves the same results as Favre. We will use the Stern-Brocot Blow-ups technique, similar to the one proposed by J. Hubbard and P. Papadopol, 2000 and 2008, to construct a compact space _ , containing (^)^2 as a dense subset, such that _ extends to a map _ _M _M as a dynamic system. We hope this method offers a more intuitive and straightforward perspective on the problem.