In this review, we apply the method of backward iteration to chaotic maps of the interval to illustrate how both the f(α) spectrum and its underlying statistical mechanics follow directly from the dynamics in statistical equilibrium. The sizes of intervals in the coarse-grained phase space are expressed directly in terms of finite-time average Liapunov exponents, representing the reverse of the information flow that is the underlying cause of deterministic chaos. The transfer matrix formulation follows directly from the method of backward iteration when addresses generated from symbolic dynamics are assigned to tree-branches. The inverse temperature is interpreted in terms of classes of initial data of the dynamical system. Finally, the usefulness of the thermodynamic formalism is illustrated by showing how the pore distribution of sandstone can be modeled by a certain two-scale Cantor set on an octal tree.