Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models

Abstract

There is a well-known correspondence between the dynamics of symplectic twist maps which represent an important class of Hamiltonian systems and the equilibrium states of a class of variational problems in solid-state physics, known as Frenkel-Kontorova models. In this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent. This allows one to transfer many ideas between the two and hence to deduce new results. For example, we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so-called non-degenerate anti-integrable limits by Aubry and Abramovici, using the concept of phonon gap and deduce that they have measure zero; we prove existence of many multi-defect equilibrium states for a Frenkel-Kontorova system if it has a single non-degenerate defect equilibrium state, via standard results in the theory of uniform hyperbolicity.