Interface free energy is the contribution to the free energy of a system due to the presence of an interface separating two coexisting phases at equilibrium. It is also called surface tension. The content of the paper is 1) the definition of the inte
rface free energy from first principles of statistical mechanics; 2) a detailed exposition of its basic properties. We consider lattice models with short range interactions, like the Ising model. A nice feature of lattice models is that the interface free energy is anisotropic so that some results are pertinent to the case of a crystal in equilibrium with its vapor. The results of section 2 hold in full generality.
We consider the map $T_{alpha,beta}(x):= beta x + alpha mod 1$, which admits a unique probability measure of maximal entropy $mu_{alpha,beta}$. For $x in [0,1]$, we show that the orbit of $x$ is $mu_{alpha,beta}$-normal for almost all $(alpha,beta)in
[0,1)times(1,infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)times(1,infty)$ along them the orbit of $x=0$ is at most at one point $mu_{alpha,beta}$-normal. These curves are disjoint and they fill the set $[0,1)times(1,infty)$. We also study the generalized $beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $beta$.
We give an algorithm, based on the $phi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is the solve an inverse problem for the dynamical systems $beta x+alpha mod1$.The first part is an expositio
n of the $phi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $phi$-expansion, necessary and sufficient conditions, which are different from those in Parrys paper.
