Resonant Equilibrium configurations in quasi-periodic media: perturbative expansions

We consider 1-D quasi-periodic Frenkel-Kontorova models. We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions to all orders for the quasi-periodic equilibria with resonant frequencies. Under very general conditions, we show that there are at least two such perturbative expansions for equilibria for small values of the parameter. We also develop a dynamical interpretation of the equilibria in these quasi-periodic media. We show that equilibria are orbits of a dynamical system which has very unusual properties. We obtain results on the Lyapunov exponents of the dynamical systems, i.e. the phonon gap of the resonant quasi-periodic equilibria. We show that the equilibria can be pinned even if the gap is zero.Comment: 19 page

A continuous family of equilibria in ferromagnetic media are ground states

We show that a foliation of equilibria (a continuous family of equilibria whose graph covers all the configuration space) in ferromagnetic models are ground states. The result we prove is very general, and it applies to models with long range interactions and many body. As an application, we consider several models of networks of interacting particles including models of Frenkel-Kontorova type on $\mathbb{Z}^d$ and one-dimensional quasi-periodic media. The result above is an analogue of several results in the calculus variations (fields of extremals) and in PDE's. Since the models we consider are discrete and long range, new proofs need to be given. We also note that the main hypothesis of our result (the existence of foliations of equilibria) is the conclusion (using KAM theory) of several recent papers. Hence, we obtain that the KAM solutions recently established are minimizers when the interaction is ferromagnetic (and transitive).Comment: 18 page

Diophantine inheritance for p-adic measures

In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a stronger form), a conjecture of Kleinbock and Tomanov \cite{KT}, as well as a question of Kleinbock \cite{Kleinbock-exponent}. Our main innovation is the introduction of a new $p$-adic Diophantine exponent which is better suited to homogeneous dynamics, and which we show to be closely related to the exponent considered by Kleinbock and Tomanov.Comment: We have split version 1 into two parts. The present paper addresses the question of inheritance of Diophantine exponents. The multiplicative case and 0-1 dichotomy will appear in a separate pape

On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics. In the language of geodesics, Almgren-Federer example constructs metrics in $\mathbb{S}^1\times \mathbb{S}^2$, with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class. In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in $\mathbb{T}^3$ for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers. For dynamics, the example also illustrates different definitions of `integrable' and clarifies the relation between minimization and hyperbolicity and its interaction with topology.Comment: 28 pages, 1 figur

Resonant equilibrium configurations in quasi-periodic media: KAM theory

We develop an a-posteriori KAM theory for the equilibrium equations for quasi-periodic solutions in a quasi-periodic Frenkel-Kontorova model when the frequency of the solutions resonates with the frequencies of the substratum. The KAM theory we develop is very different both in the methods and in the conclusions from the more customary KAM theory for Hamiltonian systems or from the KAM theory in quasi-periodic media for solutions with frequencies which are Diophantine with respect to the frequencies of the media. The main difficulty is that we cannot use transformations (as in the Hamiltonian case) nor Ward identities (as in the case of frequencies Diophantine with those of the media). The technique we use is to add an extra equation that ensures the linearization of the equilibrium equation factorizes. To solve the extra equation requires an extra counterterm. We compare this phenomenon with other phenomena in KAM theory. It seems that this technique could be used in several other problems. As a conclusion, we obtain that the perturbation expansions developed in the previous paper \cite{SuZL15} converge when the potentials are in a codimension one manifold in a space of potentials. The method of proof also leads to efficient (low storage requirements and low operation count) algorithms to compute the quasi-periodic solutions.Comment: 32 page

Calibrated configurations for Frenkel-Kontorova type models in almost-periodic environments

The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we show the existence of these configurations for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure, for any environment, the existence of calibrated configurations when the underlying dynamics is uniquely ergodic. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry-Mather theory and spaces of Delone set