Dynamics of polynomial maps on C² whose all unbounded orbits converge to one point

Abstract

In this paper, we study a family of iteration of polynomial map on the 2-dimensional complex Euclidean space C² whose all unbounded orbits converge to one point of the line at infinity in the 2-dimensional complex projective space P². In particular, we show some sufficient condition for the Lebesgue measure of its Julia set to be equal to 0.