On the dynamics near infinity of some polynomial mappings in ℂ²

2004/07/27 by Tien-Cuong Dinh, Tien‐Cuong Dinh, Dinh, Tien-Cuong +4
Mathematics · #Advanced Differential Equations and Dynamical Systems #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.CV #math.DS #msc:37Fxx

paper · pdf · doi:10.48550/arxiv.math/0407451

arxiv created 2004/07/27 · arxiv updated 2009/12/01

Abstract

We construct the Green current for a random iteration of "horizontal-like" mappings in two complex dimensions. This is applied to the study of a polynomial map f:ℂ²→ℂ² with the following properties: 1. infinity is f-attracting, 2. f contracts the line at infinity to a point not in the indeterminacy set. Then the Green current of f can be decomposed into pieces associated with an itinerary determined by the indeterminacy points. We also study the set of escape rates near infinity, i.e. the possible values of the function \limsup (1)/(n)log⁺log⁺ \normfⁿ. We exhibit examples for which this set contains an interval.

Related

• Dynamics semi-conjugated to a subshift for some polynomial mappings in C²
• Dynamics of horizontal-like maps in higher dimension
• Eigenvaluations
• Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping
• Geometry of currents, intersection theory and dynamics of horizontal-like maps