Chaotic dynamics at the boundary of a basin of attraction via non-transversal intersections for a non-global smooth diffeomorphism (Q6624014)

The paper deals with a family of maps \({T_d:\mathbb{R}^2\to\mathbb{R}^2}\), with odd \({d\geq3}\), defined as \N\[\NT_d(x,y)=(y-(x+y)^d, y-2(x+y)^d).\N\]\N\(T_d\) is a global homeomorphism, but it is not a diffeomorphism since the inverse map is not smooth over the line \({y = x}.\) The origin \((0,0)\) is a fixed point of \(T_d\) with nonempty basin of attraction \N\[ \NA_d(0)=\{(x, y)\in\mathbb{R}^2\ | \ T^n_d(x, y)\to(0,0)\ \ \text{as}\ \ \ n\to\infty\}.\N\]\NLet \(p_0=(0, 1)\) and \(p_1=(0,-1).\) The authors prove that \(\{p_0, p_1\}\) is the hyperbolic two cycle lying in the boundary of \({\partial A_d(0)}\) and the stable and unstable manifolds of \(p_1\) (as well as \(p_0\)), as a fixed point for \(T^2_d\), intersect transversally.\N\NThe main result states that there exists an invariant Cantor set, contained in \(\partial A_d(0),\) where the dynamics of \(T^2_d\) is conjugate to the full shift \({(\Sigma_N,\sigma)}.\)