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
$\mu_1 = \frac12(\delta_\frac13 + \delta_\frac23)$
and
$\mu_2 = \frac14(\delta_\frac15 + \delta_\frac25 + \delta_\frac45 + \delta_\frac35).$
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.