Symbolic extensions and dominated splittings for generic \(C^1\)-diffeomorphisms (Q2435102)

Let \(M\) be a compact boundaryless \(d\)-dimensional Riemmanian manifold, \(d\geq2\), and \(f:M\to M\) be a diffeomorphism. The authors show that for \(C^1\)-generic \(f\), if \(H_f(p)\) is the homoclinic class associated with a hyperbolic periodic point \(p,\) then either \(H_f(p)\) admits a dominated splitting of the form \(E^s\oplus F_1\oplus \cdots\oplus F_k\oplus E^u\), where \(F_i\) is not hyperbolic and onedimensional, or \(f|_{H_f(p)}\) has no symbolic extension.