I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:


Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset \Omega^{\mathbb Z}$ be an irreducible Markov shift (i.e. an irreducible 1-step subshift of finite type). Denote by $(\Sigma,\mathcal B)$ the measure space with the usual product $\sigma$-algebra.
Further, let $\mu$ be an ergodic, shift-invariant probability measure with full support on $(\Sigma,\mathcal B)$.


Question: Is $\mu$ necessarily a Markov measure?
**More precisely:** Is $\mu$ a measure that takes the value $p\_{\omega\_0} \prod\_{t=1}^{\tau}\Pi\_{\omega\_{t},\omega\_{t+1}}$ on the cylinder sets specified by the finite string $(\omega\_0,\omega\_1,\dots,\omega\_\tau)$, where $\Pi\_{i,j}$ are the elements of a compatible stochastic matrix and $p$ is its unique unity stochastic eigenvector?


It is quite easy to construct counter-examples if I drop certain assumptions. However, in the described set-up I could not find one. It feels like that question should have an easy answer, but somehow I don't get it.


Any help would be really appreciated!


Cheers,
Bernhard



Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure?
Not at all. The simplest example is provided by the so-called **$d$-Markov measures**. These are Markov measures for the associated shift whose alphabet is the subset of $\Omega^d$ which consists of all $d$-tuples of symbols that occur in $\Sigma$. It is easy to see that for $d>1$ there are more $d$-Markov measures than plain Markov ones.


If you are not satisfied with this example, then you should look up the notion of a **Gibbs measure**, see the chapter on symbolic dynamics from the lecture notes of Rufus Bowen (Springer LN, vol.470). 


