Invariant Elliptic Curves as Attractors in the Projective Plane

Let f be a rational self-map of ℙ2 which leaves invariant an elliptic curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}$\end{document} with strictly negative transverse Lyapunov exponent. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}$\end{document} is an attractor, i.e. it possesses a dense orbit and its basin has strictly positive measure.