  We study representations of the Cuntz algebras $\O_N$. While, for fixed $N$,
the set of equivalence classes of representations of $\O_N$ is known not to
have a Borel cross section, there are various subclasses of representations
which can be classified. We study monic representations of $\O_N$, that have a
cyclic vector for the canonical abelian subalgebra. We show that $\O_N$ has a
certain universal representation which contains all positive monic
representations. A large class of examples of monic representations is based on
Markov measures. We classify them and as a consequence we obtain that different
parameters yield mutually singular Markov measure, extending the classical
result of Kakutani. The monic representations based on the Kakutani measures
are exactly the ones that have a one-dimensional cyclic $S_i^*$-invariant
space.
