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Something I do not understand. It is Theorem 2.1 of the article of K.R. Parthasarathy "On the category of ergodic measures, Illinois J. Math. 5 (1961), pages 648-656 Full text here: (projecteuclid link).

Does this theorem states, as a particular case, that for any self-homeomorphism of a compact metric space, the set of ergodic invariant measures is dense in the set of all invariant measures?

If not, please can someone explain me what my mistake is in trying to apply that theorem to deduce the above statement?

If yes, please can you explain how to approximate with ergodic measures the following invariant measure in the example below?

Example in the two-dimensional torus $T: (x,y) \rightarrow (x+y,y)$; $y_1 \neq y_2$ are irrational. So $T$, if restricted to the circle $y= y_1$ and if restricted to the circle $y=y_2$, is irrational rotations. Hence the Lebesgue measures $\mu_1$ and $\mu_2$ along each of both circles, are ergodic. Take
$\mu = (\mu_1 + \mu_2)/2$. It is non-ergodic and invariant, and it seems to me that it can not be approached by ergodic measures.

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    $\begingroup$ Theorem 2.1 says that the set of ergodic measures is a G$_\delta$ in the space of probability measures (that is, an intersection of open sets)---it does not say that it is a dense G$_\delta$. There are of course many interesting self-homeomorphisms of compact metric spaces with any specified finite number of ergodic measures. $\endgroup$
    – David Handelman
    Commented Apr 14, 2017 at 16:55
  • $\begingroup$ Dear David: Thank you very much. Clearly, my mistake was to believe that a $G_{\delta}$ set denoted a set that contains a countable intersection of open and dense sets (in a Baire Space). $\endgroup$
    – Eleonora Catsigeras
    Commented Apr 14, 2017 at 17:44

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Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

A simple example of a compact space on which the ergodic measures are not dense is given by $x\mapsto x^2$ on $[0,1]$ for which the invariant measures are of the form $p\delta_0 +(1-p)\delta_1$, $p\in [0,1]$, whereas the ergodic ones are just $\delta_0$ and $\delta_1$.

You are perhaps confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, for example if the ambient space is a complete metric space.

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    $\begingroup$ I believe the OP gave a clear example of a system in which the ergodic measures were not dense in the set of invariant measures herself. $\endgroup$
    – Anthony Quas
    Commented Apr 14, 2017 at 17:21
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    $\begingroup$ Yes, I provided also a somewhat simpler example. $\endgroup$
    – coudy
    Commented Apr 14, 2017 at 17:57

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