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 $\phi\colon [0,1]\to \RR$ 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. 1 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$\phi\colon [0,1]\to \RR$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.