TITLE: What do singular, atomless invariant measures of $\times d$ look like?
QUESTION [7 upvotes]: Consider the circle map $\times d:x\mapsto dx \mod 1$. The lebesgue measure is the only absolutely continuous invariant probability measure, but this map has many other invariant measures. Of course, one can take barycentric combinations of invariant measures to get a new one, so let us restrict to the extremal points, namely ergodic measures.
One can consider a uniform measure on any periodic orbit. There are also singular, atomless invariant measures. For example, the "uniform" measure on the usual middle-third Cantor set is invariant under $\times 3$. All this is pretty explicit and I'm fine with it. But I also heard about a thermodynamical formalism that yield many invariant measures; Bowen's lecture note are on my desk but it does not seem to answer my question, which is the following: what do these measures look like? What is their support? I guess that we cannot answer this for all invariant measures, but maybe for some of them less trivial than the atomic and easy Cantor ones.
Other question: Cantor measures are easily constructed for all $\times d$, $d>2$, but I cannot really get one for $d=2$. Am I clumsy or is there something special to this case?

REPLY [3 votes]: To supplement the previous answers (and perhaps further illustrate that there are a lot of invariant measures!) I thought I'd mention what "typical" invariant measures for the $d$-fold expanding map look like. Fix a $d$-fold expanding map $T$, and consider the set $\mathcal{M}$ of all $T$-invariant Borel probability measures on the circle. We can make $\mathcal{M}$ into a compact, metrisable topological space using the weak topology, which is characterised by the fact that a sequence of measures $(\mu_n)$ converges to $\mu$ if and only if the real sequence $(\int f d\mu_n)$ converges to $\int fd\mu$ for every continuous function $f$ from the circle to the reals. We can then ask what typical elements of $\mathcal{M}$ look like in the sense of Baire category.
It turns out that the answer is this: there is a dense $G_\delta$ subset of $\mathcal{M}$ in which every measure is fully supported, weak-mixing for $T$, but not strong-mixing for $T$. All of these measures are non-atomic and pairwise mutually singular with respect to one another. So there are enormously many fully supported invariant measures - so many that it's not easy to say anything about them at all which carries with very much generality.
This result is basically due to K. R. Parthasarathy in the 1961 paper "On the category of ergodic measures". In that paper it's proved for the two-sided full shift on $d$ symbols (or even infinitely many symbols) but the same proofs go through with little change.