Introduction

Generally speaking, any physical system that can have periodic boundary conditions can also be considered in an almost periodic setup. Almost periodicity allows averaging and yields global translational invariant quantities. While this is also possible in more general ergodic setups, almost periodicity has the feature of strict ergodicity so that the mean is defined by a single configuration alone. The hull, the closure of all translates of an almost periodic configuration, is a compact topological group on which averaging with respect to the unique Haar measure gives macroscopic quantities. Almost periodic configurations can have more than the obvious translation symmetry because the dimension of the hull of the function is in general larger than the space on which the functions are defined. The other modes of symmetry are internal symmetries of the system.

Almost periodicity appears in the literature mostly for systems with an almost periodic time-dependent forcing term or when almost periodic solutions of the systems are considered. This is not the topic here. We consider particle configurations which have spatial almost periodicity and evolve them with a time-independent law. Actually, we are interested in situations where the complexity grows during the evolution, making it impossible that the solution is also almost periodic in time.

In section 2, we reconsider almost periodic lattice gas automata, for which almost periodicity is understood in the sense of minimality. In section 3, we look at spatial Bohr almost periodic Vlasov systems in which context spatial almost periodicity appears to be new. In both situations, the time evolution renders the configuration increasingly hard to describe and there is a measure for the growth of complexity.

First, we mention three other spatially almost periodic dynamical systems, which belong to the same type we are interested in but which we do not discuss any further in this paper.

1. Almost periodic KdV and Toda systems. The almost periodic Korteweg-de Vries (KdV) equation introduced in [12] is an example where spatial almost periodicity appears in fluid dynamics. It is defined as the isospectral deformation of a one-dimensional almost periodic Schrödinger operator , where V is a Bohr almost periodic function. It is a model for an almost periodic fluid in a shallow water channel. If V is periodic, one obtains the KdV equation with periodic boundary conditions as a special case. A discrete version of the almost periodic KdV is the almost periodic Toda system [7]

on the space of continuous periodic functions f. It describes a chain of oscillators with position which are nearest neighbor coupled by an exponential potential:

If is rational, then and the system is the periodic Toda lattice. It can be integrated by conjugating the flow to a linear flow on the Jacobi variety of a Riemann surface defined by the isospectrally deformed Jacobi matrix. In the almost periodic case, one still has an isospectral deformation of an almost periodic Jacobi matrix. However, an infinite dimensional generalization of the algebro-geometric integration is expected to work only in very special cases.

2. Almost periodic discrete parabolic or hyperbolic PDEs. Coupled map lattices are dynamical systems which are used to model partial differential equations (PDEs). They can be viewed as "CA with a continuum alphabet. " Such systems appear in numerical codes for PDEs. They became popular especially with the work of Kaneko, Bunimovich, and Sinai because they provide systems with "space-time chaos" which has an easy proof using an idea of Aubry [6]. An almost periodic example of a coupled map lattice is the evolution on the space of continuous functions on the d-dimensional torus, where are translations. The aperiodic configuration with evolves in time according to . The name "coupled map lattice" comes from the fact that for , the system is an array of decoupled maps , which become coupled when . Again, if all are rational, these configurations are periodic, if is irrational the configurations are almost periodic. Not only parabolic PDEs but also hyperbolic PDEs such as nonlinear wave equations, have discrete analogues as symplectic coupled map lattices. An example is on pairs of periodic functions on the torus. This discrete PDE can be rewritten as with . This is a discrete version of a nonlinear wave equation because satisfies the discrete nonlinear wave equation , a discretization . For , it is an array of decoupled Henon type twist maps. For a cubic polynomial W, it occurs as the Euler equations of a natural functional [8].

3. Almost periodic Riemannian geometry and Vlasov-Einstein dynamics. Almost periodicity is also interesting in a differential geometric setup, where interacting particles move along geodesics. An almost periodic metric g on Euclidean space defines an almost periodic Riemannian manifold on which one can average. In some sense, such a manifold looks like a torus because the mean of the curvature gives zero. Averaging through almost periodicity could be interesting in general relativity, because the Hilbert action is still defined by an almost periodic mean. Without a compactness assumption on the manifold, this would only be possible by assuming asymptotic flatness of the metric. A metric solving the almost periodic Einstein equations is a critical point of a well defined variational problem. The classic Vlasov equation considered here has as a relativistic analogue the Vlasov-Einstein equation , which describes matter not interacting through a potential but through the metric: the connection is determined from a metric g solving the Einstein equations . Existence results are only known in very special situations. Solving the Einstein equations in the almost periodic case and a better understanding of the geodesic flow in an almost periodic metric are problems that have not yet been addressed.