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Explore the dynamics of holomorphic endomorphisms in complex two-dimensional space through this 54-minute mathematical conference talk. Delve into work-in-progress research examining maps that are tangent to the identity at the origin and discover how their dynamics evolve under perturbations. Learn about the generalization of Bianchi's results and understand statements analogous to Lavaurs' theorem when unperturbed maps possess parabolic basins centered in characteristic directions without fixing complex lines. Begin with foundational motivation and results from one-dimensional cases before advancing to the more complex two-dimensional setting. Gain insights into this specialized area of complex dynamics through research conducted in collaboration with Matthieu Astorg and Lorena Lopez-Hernanz, presented during the thematic meeting on Complex Geometry, Complex Analysis and Dynamics at the Centre International de Rencontres Mathématiques in Marseille, France.
