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.
The answer that you want, namely div $U_g$, is not going to expressible in terms of a geometrically invariant quantity (such as, say, the scalar curvature of $g$) because it depends on the underlying coordinates $x$ in which you have presented the metric. For example, if $\det g(x)$ is constant (which is a coordinate dependent thing), then the divergence of $U_g$ (computed in the $xu$-coordinates, which, I assume, is what you mean by the 'Euclidean divergence') will vanish identically (and conversely, as a matter of fact).
I'll try to explain this in the symplectic formulation, since that's the version I find the clearest, but I think that you can make the translation on your own. You start with a Riemannian metric $g = dx\cdot G(x) dx$, where $G$ is a function on $\mathbb{R}^n$ with values in positive definite symmetric matrices. Let's write $G = F^TF$, where $F$ is invertible (but not necessarily symmetric; you can take $F$ to be the positive definite square root of $G$ if you like, but that's not necessary for my argument). The Lagrangian is then $L = \tfrac12 u\cdot u$, where $u = F(x)\ dx$, regarded as an $\mathbb{R}^n$-valued function on the tangent bundle of $\mathbb{R}^n$ (i.e., an orthonormal coframing of the underlying manifold). Applying the Legendre tranform, the symplectic form on the cotangent bundle pulls back to the tangent bundle to be
$$
\Omega = dp\wedge dx = d((F(x)\ u)^T) \wedge dx
= (du)^T\ F(x)^T \wedge dx + u^T\ d(F(x)^T)\wedge dx
$$
The vector field $U_g$ is the $\Omega$-Hamiltonian vector field associated to $L$, i.e., it satisfies
$$
\iota(U_g)\ \Omega = -dL = - u\cdot du
$$
(where $\iota(X)$ denotes interior product, what I normally call 'lefthook'). By the standard identity, the divergence of $U_g$ with respect to the Liouville volume form, i.e., $\mu = \tfrac1{n!}\Omega^n$, vanishes identically. Now, you can compare the Liouville volume form with the 'Euclidean' volume form in $xu$-coordinates by noting that, by exterior algebra, we have
$$
\tfrac1{n!}\Omega^n = \det(F(x))\ \tfrac1{n!}\bigl((du)^T\wedge dx\bigr)^n
$$
It follows that the 'Euclidean' divergence of $U_g$ in the $xu$-coordinates is given by the formula
$$
-U_g\bigl(\log|\det(F(x))|\bigr) = -\tfrac12\ U_g\bigl(\log|\det(G(x))|\bigr).
$$
Since, as you have already computed, $U_g(x) = F^{-1} u$, it follows that the divergence you want is, as a function, linear in the $u$-coordinates. You might write it as
$$
-\tfrac12 \nabla \bigl(\log|\det(G(x))|\bigr)\cdot (F^{-1} u).
$$
Best Answer
There are lots of zero entropy invariant probability measures, many more than just the obvious ones supported on periodic orbits. As you suggest in the question, one can understand the general case by just considering what happens for symbolic systems.
Explicit example: Let $\alpha$ be irrational and let $a_n$ denote the fractional part of $n\alpha$. Consider the sequence in $\Sigma_2 = \{0,1\}^\mathbb{Z}$ given by $x_n = 0$ if $0 \leq a_n < 1/2$, and $x_n=1$ if $1/2\leq a_n<1$. Let $X\subset \Sigma_2$ be the orbit closure of $x=(x_n)$; then there is an entropy-preserving isomorphism between the space of invariant measures for the shift map $\sigma\colon X\to X$ and for the irrational rotation $R_\alpha\colon S^1 \to S^1$. The latter preserves Lebesgue measure on the circle and is uniquely ergodic with zero entropy, so $X$ supports exactly one invariant probability measure $\mu$, which comes from Lebesgue and has zero entropy. Now $\mu$ is a shift-invariant probability measure on $\Sigma_2$ that has zero entropy but is not supported on a periodic orbit.
General result: In fact, the above construction is representative of a general phenomenon. As RW points out in his answer, you can get lots of zero entropy measures on shift spaces by taking generating partitions for zero entropy transformations. You can even get more, using the Jewett-Krieger embedding theorem (see Petersen's "Ergodic Theory" or Denker, Grillenberger, and Sigmund's "Ergodic Theory on Compact Spaces"), which lets you find a closed shift-invariant subset of the shift space that has the desired measure as its only shift-invariant probability measure. So there's a lot there.