On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$

Tien-Cuong Dinh; Romain Dujardin; Nessim Sibony

On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$

Dynamical Systems 2007-05-23 v1 Complex Variables

Authors: Tien-Cuong Dinh , Romain Dujardin , Nessim Sibony

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 $f:\mathbb{C}^2\to\mathbb{C}^2$ 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 $\limsup \frac{1}{n}\log^+\log^+ \norm{f^n}$ . We exhibit examples for which this set contains an interval.

Keywords

polynomial maps green current random iteration escape rates indeterminacy points

Cite

@article{arxiv.math/0407451,
  title  = {On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$},
  author = {Tien-Cuong Dinh and Romain Dujardin and Nessim Sibony},
  journal= {arXiv preprint arXiv:math/0407451},
  year   = {2007}
}

On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$

Abstract

Keywords

Cite

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