This question is inspired by the construction of the time evolution for endomotives as given by Connes and Marcolli in their book http://www.alainconnes.org/docs/bookwebfinal.pdf.

Let $M$ be a monoid (countable and discrete) acting on a locally compact Hausdorff space $X$ and consider the $C^*$-algebra $A$ given by the (semigroup) crossed product $$A = C_0(X) \rtimes M$$ Now let us assume given a state $\mu : A \to \mathbb C$ (for example one could try to use the one induced by the integration map $f \in C_0(X) \mapsto \int_X f $ with respect to the Haar measure on $X$, as in the above reference) such that the associated GNS constrution yields a faithful representation $\pi : A \to \mathcal B (H)$, further let us assume that the map $\mathbb R \to Aut(\pi(A)'')$ induced by Tomita-Takesaki theory on the von Neumann algebra given by the bicommutant $\pi(A)''$ restricts to a map $$\sigma : \mathbb R \to Aut(A)$$ My question is now if there are known examples where $\sigma$ is described explicitly in the literature. (Of course my general formulation allows trivial examples which are not asked for...)

In some sense $\sigma$ should depend only on $M$ and its action on $X$ because this is the only source for noncommutativity. Recall that Tomita-Takesaki theory is only visible in the noncommutative case, in the commutative case the $\mathbb R$-action is trivial.

The only cases I know are essentially all given by so called "Bost-Connes type systems", as explained for example in the nice article http://arxiv.org/pdf/0710.3452v2 by Laca, Larsen and Neshveyev. (The best known example is given by the original Bost-Connes system $C(\hat {\mathbb Z}) \rtimes \mathbb N$.)