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.