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Definition: Let $X$ be a compact metric space and let $\mu$ be a Borel probability measure on $X$, then we say that the sequence $(x_n)$ of elements of $X$ is equidistributed with respect to $\mu$ if we have that for any $f \in C(X)$ $$\frac1n \sum_{m = 1}^n f(x_m) \to \int_X f(x) \, d\mu(x).$$

Now, I want to find a non-ergodic invariant measure such that $\{2^n x\} = 2^n x \text{ mod } 1$ is equidistributed with respect to this measure for some $x$ (then $x$ is said to be a generic point). I want to find this $x$.

So, I know that ergodic-measures are extreme points of the space of invariant measures so if I take two ergodic ones (which should not be so hard to find) and take a convex combination I would have an example. Fine. I can note that $\frac13$ and $\frac15$ are periodic with period 2 and 4 respectively. So I take the ergodic measures

$$\mu_1 = \frac12(\delta_\frac13 + \delta_\frac23)$$


$$\mu_2 = \frac14(\delta_\frac15 + \delta_\frac25 + \delta_\frac45 + \delta_\frac35).$$

So now a convex combination of $\mu_1$ and $\mu_2$ should give me an example of a non-ergodic measure (the ergodic measures are dense in the space of invariant measures).

I know that $x = \frac13$ and $x = \frac15$ are generic points for $\mu_1$ and $\mu_2$ respectively. But how do I find a generic point for a convex combination of $\mu_1$ and $\mu_2$?

Am I on the right track? Please only a hint, this is homework.

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up vote 3 down vote accepted

Mixing the binary expansion of $1/3 = .01010101\ldots$ and $1/5 = .001100110011\ldots$ should give you such a generic point :

For example, $x = .01 \;\; 0011 \;\; 010101 \;\; 00110011 \;\; 0101010101 \;\; 001100110011 \ldots$ is a generic point for $(\mu_1 + \mu_2)/2$

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