Ergodicity and constant Lyapunov exponents on attractors

Question

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?

Answer 1

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?

For every definition of attractor that I am aware of, an attractor is minimal and inseparable, i.e., it does not have any true subset that is an attractor itself. If two trajectories starting on an attractor would visit disjoint sets of points, these sets would be separate attractors. Now, once two trajectories have visited the same point, they are identical except for a time shift on account of the system being deterministic.

However there is some fine print to be had here, namely what exactly do we mean by attractor, inseparable, after transients, visit, equal frequency, etc. For any non-pathologic system, it doesn’t matter which exact definition of ergodic or attractor you use, however, there are systems exhibiting weak ergodicity breaking (see, e.g., this paper). The phase space of these systems can be segmented into regions that are connected, but the expected waiting time for going from one region to another is infinite. Depending on your definitions, all the regions form one attractor, but the system is not ergodic (even if on the attractor).

Finally, even for not-pathologic system, you need to pay attention what exactly you mean with equal frequency, as the attractor is not equally dense everywhere in phase space – but then the attractor has measure zero with respect to a measure of the phase space.