Homoclinic tangencies in \(\mathbb{R}^n\)
The celebrated theorem of Birkhoff-Smale provides a method for rigorously concluding the existence of a horseshoe structure. The basic assumption of this theorem is the presence of a transverse homoclinic point. The assumption of transversality is not easy to verify for concrete dynamical systems. This paper is one of many attempts to remove this assumption in the following situation: Let \(M\) be an \(n\)-dimensional smooth manifold, and let \(f: M\to M\) be a diffeomorphism with a hyperbolic fixed-point \(p\in M\) and with stable and unstable invariant manifolds \(W^s\) and \(W^u\) at \(p\). The author proves that if \(f\) is locally \(C^1\)-linearizable in a neighborhood of \(p\), \(W^u\) and \(W^s\) have complementary dimensions \(n-1\) and \(1\), \(W^u\) and \(W^s\) have a finite tangential contact at an isolated homoclinic point \(B\) and \(W^u\) contains a graph portion \(\Lambda\) associated with \(B\), then the invariant manifolds have a point of transversal intersection arbitrary close to the point \(B\). Therefore, \(f\) has a horseshoe. The paper is well written and organized.
scientific article; zbMATH DE number 2155032