Transversal homoclinic points of the Hénon map (Q2504896)

The authors study the existence of transversal homoclinic points for the Hénon map \(F(x,y)=(a-x^2+by, x),\) for the interesting parameters \(|b|\leq1,\) \(a>-(1-b)^2/4.\) They prove that for \(|b|\in(0,1]\) and \(a\geq 20+91b/20-19b^2/400\) these homoclinic points exist. For the particular case \(b=-1\) this result is improved showing that the transversal homoclinic point also exists for \(a\geq17/64.\) The main technique used in the proofs is a shadowing theorem developed in previous papers. This result allows to show the existence of a homoclinic hyperbolic orbit which, by using previous results of K.J. Palmer, turns out to be transversal. The authors also announce that by using a computer-assisted approach the results for \(b=-1\) can be extended until \(a\geq-0.866360\ldots,\) in support of an old conjecture of Devaney and Nitecki that claims the same result for \(a>-1,\) but no details are given in this paper.