Journal of Statistical Physics, Vol. 110, Nos. 1/2, January 2003. p. 73-86.
We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as ‘‘spin chains,’’ with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle–Perron–Frobenius operator), which is defined using the maps (presentation functions) generating the Farey ‘‘tree.’’ The spectrum of this operator was completely determined by Prellberg...