Generic uniqueness of minimal configurations with rational rotation numbers in Aubry-Mather theory (Q1773547)

This paper considers so-called \((h)\)-minimal configurations in Aubry-Mather theory. A configuration is a bi-infinite sequence \(\{x_i\}\) where \(i\in\mathbb{Z}\) and \(\chi_i\in\mathbb{R}\). The function \(h:\mathbb{R}^2\to\mathbb{R}^1\) is extended to arbitrary finite sequences \(\{\chi_j,\dots,\chi_k\}\) \((j<k)\) of configurations by \[ h(\chi_j,\dots,\chi_k)=\sum^{k-1}_{i=j}h(\chi_i,\chi_{i+1}). \] A segment is \((h)\)-minimal if \(h(\chi_j,\dots,\chi_k)\leq h(y_j,\dots,y_k)\) whenever \(\chi_j=y_j\) and \(\chi_k=y_k\). The author assumes the following properties of \(h\): (i) For all \((\xi,\eta)\in\mathbb{R}^2\), \(h(\xi-\eta)\); (ii) \(\lim_{|\eta|\to\infty}h(\xi,\xi+\eta)=\infty\) uniformly in \(\xi\); (iii) If \(\xi+1<\xi_2\) and \(\eta_1<\eta_2\), then \(h(\xi_1,\eta_1)+h(\xi_2,\eta_2)<h(\xi_1,\eta_2)+h(\xi_2,\eta_1)\); and (iv) If \((\chi_{-1},\chi_0,\chi)\neq (y_{-1},y_0,y_1)\) are \((h)\)-minimal segments and \(\chi_0=y_0\), then \((\chi_{-1}-y_{-1})(\chi_1-y_1)<0\). Finally, a configuration \(\{\chi_i\}\) is \((h)\)-minimal if, for each pair of integers \(j\) and \(k\) with \(j<k\) and each finite segment \(\{y_i\}^k_{i=j}\subset \mathbb{R}^1\) satisfying \(\chi_j=y_j\) and \(\chi_k=y_k\) the inequality \[ h(x_j,\dots,\chi_k)\leq h(y_j,\dots,y_k) \] holds. It is known that the set \(M(h)\) of all \((h)\)-minimal functions is closed. The author establishes the existence of a set of functions, a countable intersection of open dense subsets, such that for each rational number \(\alpha\), the following properties hold: (1) There exist three different \((h)\)-minimal configurations with rotation number \(\alpha\); and (2) Any \((h)\)-minimal configuration with rotation number \(\alpha\) is a translation of one of these configurations. Given the author's extensive assumptions and the nonintuitive character of \((h)\)-minimal configurations, this paper would have benefitted greatly from some examples.