Normal Lyapunov exponents and asymptotically stable attractors

Suppose f is a C ^1+α map and leaves a lower-dimensional compact attractor A. In this article, we show that if for every f-ergodic probability measure supported on A, the normal Lyapunov exponents are negative, then this attractor could be a high-dimensional attractor. Moreover, we prove that the supremum of the normal Lyapunov exponents on the set of all ergodic measures can be achieved.