Transversal homoclinic points of the Hénon map

Using shadowing techniques we prove that the Hénon map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{a,b}(x,y)=(a-x^{2}+by,x)$\end{document} admits a transversal homoclinic point for a set of parameters which is not small. For the area and orientation preserving Hénon map (corresponding to b=-1) we prove that a transversal homoclinic point exists for a≥0.265625. Applying a computer-assisted version of our scheme we show that the result holds even for a≥-0.866. This supports an old conjecture due to Devaney and Nitecki dating back to 1979, see [4], claiming that the Hénon map in the case b=-1 admits a transversal homoclinic point for a>-1.