Monotone gradient dynamics and Mather's shadowing

Abstract

We prove that all the invariant measures for the monotone gradient dynamics of the (non-tilted) Frenkel-Kontorova model are supported on the set of stationary configurations, and that the ω-limit set of each configuration is either a single stationary configuration or contains an invariant circle of stationary configurations. A consequence is a new proof of the existence of Mather's shadowing orbits of twist diffeomorphisms. We compare the results with the dynamics of other dissipative spatially extended systems, such as the real Ginzburg-Landau equation, and suggest application to non-conservative diffusion on 1D lattices.

Construction of invariant measures of Lagrangian maps: minimisation and relaxation

“…We set z = (max(x i , y i )) i∈Z (or z = (min(x i , y i )) i∈Z ). The main result of [Sli99d] is that d(z) < ∞ if and only if x and y are in the same Birkhoff region of instability.…”

Positive topological entropy for monotone recurrence relations

“…The idea of constructing two solutions of (1.1) exchanging rotation numbers is borrowed from [23], in which the gradient flow is investigated in configuration space with bounded action. We should mention that the method in [23] depends heavily on Aubry's lemma that two minimizers cross at most once, which we do not have for general monotone recurrence relations.…”

“…We should mention that the method in [23] depends heavily on Aubry's lemma that two minimizers cross at most once, which we do not have for general monotone recurrence relations.…”

“…In order to make the approach of [23] work for monotone recurrence relations, we shall first establish the following result: for monotone recurrence relation (1.1), each minimizer with bounded action is Birkhoff (Theorem 3.15). We remark that by proving such a conclusion we in fact give as a byproduct an affirmative answer to a question of Blank in the case of monotone recurrence relations; see the last paragraph of [9, §1].…”

Total destruction of invariant tori for the generalized Frenkel–Kontorova model

“…The case d = 1 corresponds to twist maps. Destruction of invariant tori for this model is closely related to the instability problems in solids, such as diffusions on lattices [Sli99] and so on.…”