Fast ergodicity of rotations on the circle and stability of one-dimensional non-periodic Sturmian ground states

Abstract

Rotations on the circle by irrational numbers give rise to uniquely ergodic Sturm dynamical systems. We show that rotations by badly approximable irrationals have the property of fast ergodicity. It was shown recently that any Sturmian ergodic measure is the unique ground state of a non-frustrated Hamiltonian of one-dimensional classical lattice-gas model. We use the fast ergodicity property to show that for slowly decaying interactions, $1/r^{\alpha}$ with $1 < \alpha < 3/2$, non-periodic Sturmian ground states are stable with respect to periodic configurations consisting of Sturmian words.