Thermodynamics of a set of DNA on trees

We define a DNA as a sequence of $\pm 1$’s and embed it on a path of Cayley tree. Using group representation of the Cayley tree, we give a hierarchy of a countable set of DNAs each of which ’lives’ on the same Cayley tree. Then we give a mode of this set of DNAs by an analogue of Ising model with three spin values (considered as DNA base pairs) on a set of admissible configurations. To study thermodynamic properties of the model of DNAs we describe corresponding translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two. We use these results to give distributions of Holliday junctions and branches of DNAs. In case of very high and very low temperatures we give stationary distributions and typical configurations of the Holliday junctions.