Symbolic extensions and dominated splittings for generic C^{1}-diffeomorphisms

Let $\mathrm{Diff }^1(M)$ be the set of all $C^1$ -diffeomorphisms $f:M\rightarrow M$ , where $M$ is a compact boundaryless d-dimensional manifold, $d\ge 2$ . We prove that there is a residual subset $\mathfrak R $ of $\mathrm{Diff }^1(M)$ such that if $f\in \mathfrak R $ and if $H(p)$ is the homoclinic class associated with a hyperbolic periodic point $p$ , then either $H(p)$ admits a dominated splitting of the form $E\oplus F_1\oplus \dots \oplus F_k\oplus G$ , where $F_i$ is not hyperbolic and one-dimensional, or $f|_{H(p)}$ has no symbolic extensions.