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?
Best Answer
If you do the Cantor measure construction for d=2, you just get Lebesgue measure... so it's a little bit special.
There are lots of fully supported invariant measures for the map $\times d$: the thermodynamic formalism that you mention gives you a whole zoo of them. In particular, if $\phi\colon [0,1]\to \mathbb{R}$ is any Hölder continuous function, then there is a unique "equilibrium state" for $\phi$, which is a probability measure $\mu_\phi$. This measure can be shown to have a certain Gibbs property, which in particular implies that it has full support -- it gives positive measure to every open set in $[0,1]$.
The easiest of these to think about are the Bernoulli measures. Given a sequence $x_1 x_2 \cdots x_n$ where $x_i \in \{0,1,\dots,d-1\}$, consider the interval $$ C(x_1 x_2 \cdots x_n) = [x_1 d^{-1} + x_2 d^{-2} + \cdots + x_n d^{-n}, x_1 d^{-1} + x_2 d^{-2} + \cdots + (x_n+1) d^{-n}]. $$ These generate the $\sigma$-algebra of Borel sets, so given any probability vector $\mathbf{p} = (p_1,\dots,p_d)$, we can define an invariant measure $\mu_{\mathbf{p}}$ by $$ \mu_{\mathbf{p}}(C(x_1 \cdots x_n)) = p_{x_1} p_{x_2} \cdots p_{x_n}. $$ These are equilibrium states for potential functions that are constant on the $d$ intervals $C(x_1)$. Potential functions that are constant on intervals $C(x_1 \cdots x_n)$ for some $n$ yield Markov measures as equilibrium states, and these can also be described quite explicitly.
All these measures are invariant under the $\times d$ map, fully supported on the interval, ergodic, and non-atomic.
See this answer for a little bit more on Markov measures, and this one for some other invariant measures that are ergodic, non-atomic, and fully supported, but have zero entropy. It's worth pointing out that pretty much anything you say about invariant measures for the shift map $\sigma\colon \Sigma_d^+ \to \Sigma_d^+$ (here $\Sigma_d^+ = \{0,\dots,d-1\}^{\mathbb{N}}$) can be translated into a statement about invariant measures for the $\times d$ map, since the two are topologically conjugate on a total probability set -- that is, a set that is given full weight by every invariant measure.