On the hyperbolicity of homoclinic classes
The authors study the hyperbolicity of a diffeomorphism \(f\) on a compact manifold. More precisely, they seek to investigate the hyperbolicity of an invariant compact set by using the hyperbolicity of the periodic points in this set. In general, the hyperbolicity of periodic orbits is not enough to conclude the hyperbolicity of the set. The main result states that if \(p\) is a hyperbolic periodic point and its homoclinic class \(H(p)\) admits a (homogeneous) dominated splitting \(T_{H(p)}M=E \oplus F\), where \(E\) is contracting and \(\text{dim}(E)=\text{ind}(p)\). If, moreover, all points homoclinically related with \(p\) are uniformly \(F\)-expanding, then \(F\) is uniformly expanding on \(H(p)\). In particular, \(H(p)\) is hyperbolic. The paper contains an introduction to the results and the proofs; the authors also derive and describe some consequences of their main result.
scientific article; zbMATH DE number 5648551