Ergodic Transport Theory, Periodic Maximizing Probabilities and the Twist Condition

Consider the shift T acting on the Bernoulli space \(\varSigma =\{ 1,2,3,..,d\}^{\mathbb{N}}\) and \(A:\varSigma \rightarrow \mathbb{R}\) a Holder potential. Denote $$\displaystyle{m(A) =\max _{\nu \mbox{ an invariant probability for $T$}}\int A(x)\;d\nu (x),}$$ and, μ ∞, A , any probability which attains the maximum value. We will assume that the maximizing probability μ ∞ is unique and has support in a periodic orbit. We denote by \(\mathbb{T}\) the left-shift acting on the space of points \((w,x) \in \{ 1,2,3,..,d\}^{\mathbb{Z}} =\varSigma \times \varSigma =\hat{\varSigma }\). For a given potential Holder \(A:\varSigma \rightarrow \mathbb{R}\), where A acts on the variable x, we say that a Holder continuous function \(W:\hat{\varSigma }\rightarrow \mathbb{R}\) is a involution kernel for A (where A ∗ acts on the variable w), if there is a Holder function \(A^{{\ast}}:\varSigma \rightarrow \mathbb{R}\), such that, $$\displaystyle{A^{{\ast}}(w) = A \circ \mathbb{T}^{-1}(w,x) + W \circ \mathbb{T}^{-1}(w,x) - W(w,x).}$$ One can also consider V ∗ the calibrated subaction for A ∗, and, the maximizing probability \(\mu _{\infty,A^{{\ast}}}\) for A ∗. The following result was obtained on a paper by Lopes et al.: for any given x ∈ Σ, it is true the relation $$\displaystyle{V (x) =\sup _{w\in \varSigma }\,[\,(W(w,x) - I^{{\ast}}(w)) - V ^{{\ast}}(w)\,],}$$ where I ∗ is non-negative lower semicontinuous function (it can attain the value ∞ in some points). In this way V and V ∗ form a dual pair. For each x one can get one (or, more than one) w x such attains the supremum above. That is, solutions of $$\displaystyle{V (x) = W(w_{x},x) - V ^{{\ast}}(w_{ x}) - I^{{\ast}}(w_{ x})\,.}$$ A pair of the form (x, w x ) is called an optimal pair. Under some technical assumptions, we show that generically on the potential A, the set of possible optimal w x , when x covers the all range of possible elements x in ∈ Σ, is finite.