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:



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> 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?
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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.



Classification of ergodic measures for circle expanding maps
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.


