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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?

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    $\begingroup$ @Ian, you might want to read the relevant sections about d-bar distance in Shields' - "The Ergodic Theory of Discrete Sample Paths", it is focused towards information theory rather than ergodic theory, but it gives some intuition, at-least in the Bernoulli case. $\endgroup$
    – Asaf
    Commented Jan 18, 2012 at 13:55
  • $\begingroup$ strictly ergodic? = uniquely ergodic and fully supported? $\endgroup$
    – Anthony Quas
    Commented Jan 18, 2012 at 16:20
  • $\begingroup$ Anthony: by saying that $\mu$ is strictly ergodic I mean that the restriction of the shift to the support of $\mu$ is uniquely ergodic. This usage occurs in some of Oliver Jenkinson's papers (e.g. Ergodic optimisation and Every 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. $\endgroup$
    – Ian Morris
    Commented Jan 18, 2012 at 17:22
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    $\begingroup$ Hi Ian, I fairly strongly suspect the answer is yes. Writing down a proof is unpleasant though. Here is a rough strategy I'd try. Take your favourite ergodic measure (think Bernoulli). It roughly has the right proportions of 1's and 0's in almost every 1000000 block. Use a Rokhlin tower to fix the density to within epsilon ( larger than the error you expect from CLT but still small) on each 1000000 block. You have to make very few changes to do this. Now work on much longer blocks etc. $\endgroup$
    – Anthony Quas
    Commented Jan 26, 2012 at 5:22
  • $\begingroup$ Thanks, Anthony. I am also wondering about some other issues, such as whether there is a nice characterisation of the d-bar closure of the set of ergodic measures supported on periodic orbits. It seems to me that the general structure of the space of measures in the d-bar topology has been studied little enough that it might be worthwhile devoting a paper to the topic, although interest in such a paper would presumably not be very high. $\endgroup$
    – Ian Morris
    Commented Jan 26, 2012 at 11:42

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The strictly ergodic measures are dense in the ergodic measures in the $\bar d$-metric. The proof (actually, a sketch of the proof) is given in Single orbit dynamics by Weiss (Theorem 4.4', page 46).

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