I am currently studying a problem which deals with cocycles of highly
noncompact operators on Hilbert space, with the base transformation
being the full shift on two symbols. In my particular situation it
turns out that the top Lyapunov exponent of a fixed cocycle,
considered as a function on the space of invariant measures, is discontinuous with respect to the weak-* topology but Lipschitz continuous with respect to Ornstein's $\overline{d}$-metric.
However I do not have very much intuition for the topology on the
invariant measures induced by $\overline{d}$, and the resources on
this seem relatively limited (the best I have found so far is Glasner's book *Ergodic theory via joinings*). I am currently trying to understand the generic features of the set of ergodic measures with respect to the $\overline{d}$-metric.

With regard to the space of shift-invariant measures under the weak-* topology, the following facts have been well-known for many years, mostly dating back to Parthasarathy's 1961 paper *On the category of ergodic measures*:

- The space of measures under the weak-* topology is a compact metrisable space (elementary, as these things go).
- Measures supported on a single periodic orbit are dense. In particular, ergodic measures are dense, and strictly ergodic measures (i.e. those whose support is uniquely ergodic) are dense (Parthasarathy 1961).
- Weak-mixing measures are a dense residual set. (Density follows from the density of strong-mixing measures; the $G_\delta$ property follows from a modification of Parthasarathy's work.)
- Strong-mixing measures are a dense meagre set. (Meagreness is due to Parthasarathy. Density of strong-mixing measures follows from density of Gibbs measures.)
- Fully-supported measures are a dense residual set. (Fairly elementary.)
- Zero-entropy measures are a dense residual set. (Follows from density of periodic orbits and semicontinuity of entropy.)

Most of these statements have known analogues in the $\overline{d}$-metric, namely:

- The space of measures under the $\overline{d}$-metric is complete but not separable.
- Measures supported on periodic orbits are not dense.
- The space of ergodic measures is closed, as are the spaces of strong-mixing and Bernoulli measures.
- Fully-supported measures are a dense residual subset of the ergodic measures.
- Entropy is continuous.

However, it is not clear to me whether or not strictly ergodic measures are dense in the ergodic measures in the $\overline{d}$-metric. Does anyone know whether or not this is the case?

Ergodic optimisationandEvery ergodic measure is uniquely maximising) but it does seem to differ slightly from earlier use by Grillenberger and Shields, where it is the support and not the measure that is called strictly ergodic. $\endgroup$