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I would like to know an example (not using the Gibbs measure Theory) of a sequence of measures $\mu_n:\mathcal B\to[0,1]$ , where $\mathcal B$ is the $\sigma$-algebra of the borelians of a compact space $X$ such that :

1) $\mu_n$ is ergodic, with respect to a fixed continuous function $T:X\to X$, for all $n\in\mathbb N$;

2) $\mu_n\to \mu$ in the weak-$*$ topology and $\mu$ is not ergodic.

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4 Answers 4

up vote 10 down vote accepted

Let $X=\{0,1\}^{\mathbb{N}}$ with the infinite product topology (which is metrisable). For each $n \geq 1$, define $x_n$ to be the sequence given by $x_i=0$ for $1 \leq i \leq n$, $x_i=1$ for $n+1 \leq i \leq 2n$, and $x_{2n+i}=x_i$ for all $i$. Let $T \colon X \to X$ be the shift transformation $T[(x_n)]= (x_{n+1})$. We have $T^{2n}x_n=x_n$ for every $n \geq 1$, so the measure $\mu_n$ defined by $\mu_n:=(2n)^{-1}\sum_{j=0}^{2n-1}\delta_{T^jx_n}$ is an ergodic invariant Borel probability measure for $T$. Let $\overline{0}$ denote the element of $X$ corresponding to an infinite sequence of zeroes, and similarly let $\overline{1}$ denote the infinite sequence of ones; we have $\lim_{n \to \infty} \mu_n = \frac{1}{2}(\delta_{\overline{0}}+\delta_{\overline{1}})$, and this limit is not ergodic (since the set containing only the point $\{\overline{0}\}$ has measure 1/2 but is invariant).

There is a nice paper by Parthasarathy - called, I think, "On the category of ergodic measures" - which shows that for this particular dynamical system and some of its generalisations, the set of all ergodic measures and the set of all non-ergodic measures are both weak-* dense in the set of all invariant measures, so this phenomenon can actually happen quite a lot.

(Hmm, the definition of $X$ above is supposed to have curly set brackets in it, but I can't get them to appear for some reason. Anyway, it's supposed to be the set of all one-sided infinite sequences of zeroes and ones.)

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Thank you very much Ian. –  Leandro May 4 '10 at 16:39
    
You're welcome =o] –  Ian Morris May 4 '10 at 19:54
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If you just want something elementary to present to students, take $X$ to be the annulus $1\le|z|\le 2$ and $T(z)=ze^{i(|z|-1)}$. Then the Lebesgue measure on every circle of radius $1+\frac 1n$ is ergodic (if the ergodicity of the irrational rotation has not been covered yet, take the counting measures on some orbits on the circles of radii close to $1$ where the rotation is rational instead) but the weak limit, which is the Lebesgue measure on the unit circle, is as far from ergodic as it can possibly be.

Of course, Ian's example is far more interesting in many respects. :-)

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fixed braces, you need to double the backslash –  Gerald Edgar May 4 '10 at 17:41
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There's a property called "entropy density of ergodic measures" (or variations on that terminology), which states that given an invariant measure μ, you can find a sequence of ergodic measures μn that converges to μ in the weak* topology, and furthermore, the lim inf of the entropies hμn(f) is at least hμ(f). In other words, not only can you approximate an arbitrary invariant measure using ergodic measures, but you can do so without losing any of the entropy of the original measure.

Of course not every system has entropy density, but there are many interesting ones that do. Pfister and Sullivan use this property in a couple papers -- see "Large deviations estimates for dynamical systems without the specification property" (Nonlinearity 18, 2005, 237-261), and "On the topological entropy of saturated sets" (Ergod. Th. & Dynam. Sys. 27, 2007, 929-956). In particular, they show that entropy density follows from something they call the g-almost product property. This latter property is a weaker form of the classical specification property (it's also been called almost specification).

There are many systems known to have specification -- for example, if an Anosov systems, an Axiom A system, a subshift of finite type, or an interval map is topologically mixing, then it has specification, and hence has entropy density. There are also classes of systems that satisfy the g-almost product property (but not specification) and hence have entropy density: β-shifts are one example.

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Hi Vaughn, the papers you are pointing out seems very interesting. I knew that property only for some classes of Gibbs measures. This class is obtained by a perturbation of a ferromagnetic Ising model by a antiferromagnetic interaction of long-range. I don't read yet carefully the references you provided, but it is possible that the example I mentioned to be a particular case of them. Thank you for the reply. –  Leandro May 19 '10 at 2:01
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consider the torus map $(x,y) \mapsto (x+y, y)$. For every $y$, this gives a rotation on the circle $S^1 \times y$. That is, Lebesgue measure on any "horizontal" circle is preserved. For $y$ rational it is not ergodic, but for $y$ irrational it is ergodic. Well, that's it. Lebesgue on a circle with rational rotation is weak* approximated by Lebesgue on circles with irrational rotation.

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