Ergodicity and constant Lyapunov exponents on attractors

The Wikipedia article on Lyapunov exponents says the following:

For a [real] dynamical system with evolution equation $$\dot{x_i} = f_i(x)$$ in an n–dimensional phase space, the spectrum of Lyapunov exponents $$\{\lambda_1, \lambda_2, \dots, \lambda_n \}$$ in general, depends on the starting point $$x_0$$. However, we will usually be interested in the attractor (or attractors) of a dynamical system, and there will normally be one set of exponents associated with each attractor. The choice of starting point may determine which attractor the system ends up on, if there is more than one.

In the same paragraph it says:

The set of Lyapunov exponents will be the same for almost all starting points of an ergodic component of the dynamical system.

Am I to conclude from this that once a system has settled on an attractor, it is always ergodic? In other words, after the transients of the system, the system will simply visit all states on the attractor with an equal frequency?