Classification of ergodic measures for circle expanding maps Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by
$$\times_d(x) = dx \mod 1$$
where the degree $d$ is any integer greater than $1$.
There are a wealth of ergodic invariant measures; I know at least of uniform measures on periodic orbits, the Lebesgue measure, Gibbs measures for Hölder potentials, Bernoulli and Markov measures.
My question is:

To which extend do we know a kind of classification of all ergodic measures of $\times_d$? For example, is there a precise classification of ergodic measure of positive entropy?

I am pretty sure no complete, very explicit classification is known, because it would probably answer Furstenberg's conjecture (Lebesgue measure is the only non-atomic measure which is invariant under both $\times_2$ and $\times_3$).
Any classification, however rough, or examples of invariant measures not cited above would be welcome.
 A: You're right. The set of measures for these maps is a zoo! There is the obvious map (base $d$ expansion) $\pi$ from $\{0,1\ldots,d-1\}^{\mathbb N}$ to $[0,1)$ which is a bijection off a countable set. $\pi$ is then a factor map from the full one-sided $d$-shift to $\times_d$. 
There are only two ergodic invariant measures that give positive measure to the set where $\pi$ fails to be 1-1, namely the $\delta$-measures at $\overline 0$ and $\overline{d-1}$. This means that you can study invariant measures for $\times_d$ by studying invariant measures for the full shift on $d$ symbols, and there is a 1-1 correspondence between these sets of measures except for the additional fixed point that I mentioned. 
There is also a natural bijection between one-sided and two-sided invariant measures on the full $d$-shift. 
Maybe a nice way to dramatize the fact that there are lots of measures is to quote the Krieger generator theorem: for any invertible ergodic process at all (say on a Lebesgue space) with entropy less than $\log d$, there exists a generator: a finite partition such that the set of points that have a `twin' (another point whose entire forward and backward orbit follows the same sequence of partition elements) has measure 0. There is then a measure-theoretic isomorphism between your arbitrarily chosen ergodic transformation with entropy less than $\log d$ and $\times_d$ equipped with an appropriate ergodic invariant measure.
This property of $\times_d$, that it contains copies of all processes with entropy less than its own topological entropy is called universality. Some other universal processes are known.
