We know that any substitution can generate sequence, for example the Fibonacci substitution:


$\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of $\sigma$. ($\sigma(\omega)=\omega$). We can define an orbit space which is based on the fixed point of the substitution, for example, the Fibonacci substitution, we can define $\Omega=\overline{\{T^{n}(\omega):n\in \mathbb{N}\}}$, here $T$ is the left shift. This system is very good as it has a uniquely ergodic measure, say $\mu$. By ergodic theorem we have that $\mu([0])=\mbox{frequency of }0 $, $\mu([1])=\mbox{frequency of }1 $, where $[0],[1]$ are cylinders in $\Omega$. Also we can calculate the measure of other cylinders, such as $\mu([01]),\mu([100])$ and so on. We can use the irrational rotation to find the formula of the measure of each block.


My question is that if given other substitution, for example, $\sigma(1)=110, \sigma(0)=11$, we can also define the orbit space according to the fixed point of this substitution. Let the orbit space be $X=\overline{\{T^{n}(\tau):n\in \mathbb{N}\}}$, where $\tau$ is the fixed point of the substitution mentioned above. We can always define an invariant measure, say $\nu$, on $X$. In fact it is the limit of $\nu\_n=\dfrac{1}{n}\sum\_{i=1}^{n-1}\delta\_{T^{i}(\tau)}$
in the weak sense. Here $\delta\_{x}$ is the Dirac measure. How can we calculate the measure of cylinders, for instance, $\nu([1])=?$
$\nu([11])=?$ Can we give a formula to find the measure of all the cylinders? 



Substitutions and Sturmian sequences
My understanding is that there are no general techniques for answering your question, even for some specific examples. Certainly this was the case a few years ago. 


A relatively well known example is the Kolakoski sequence, see <http://mathworld.wolfram.com/KolakoskiSequence.html> . This can be generated by a length two substitution on two symbols but the values of $\nu[1]$ and $\nu[0]$ are not known (they are conjectured to be 1/2).


I don't know much about the example, but I imagine that if you have access to the article 'What is the long range order in the Kolakoski sequence?' by Michel Dekking it would probably explain the problem quite neatly.


