The hydrogen identity for Laplacians

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, where H=(d+d^*)^2 is the sign-less Hodge Laplacian defined by sign-less incidence matrix d and where L is the connection Laplacian. Any spectral information about L directly leads to estimates for the Hodge Laplacian H=(d+d^*)^2 and allows to estimate the spectrum of the Kirchhoff Laplacian H_0=d^* d. The hydrogen identity implies that the random walk u(n) = L^n u with integer n solves the one-dimensional Jacobi equation Delta u=H^2 with (Delta u)(n)=u(n+2)-2 u(n)+u(n-2). Every solution is represented by such a reversible path integral. Over a finite field, we get a reversible cellular automaton. By taking products of complexes such processes can be defined over any lattice Z^r. Since L^2 and L^(-2) are isospectral, by a theorem of Kirby, the matrix L^2 is always similar to a symplectic matrix if the graph has an even number of simplices. The hydrogen relation is robust: any Schr\"odinger operator K close to H with the same support can still can be written as $K=L-L^{-1}$ where both L(x,y) and L^-1(x,y) are zero if x and y do not intersect.Comment: 29 pages, 8 figure

Cellular Automata vs. Quasisturmian Shifts

If L=Z^D and A is a finite set, then A^L is a compact space. A cellular automaton (CA) is a continuous transformation F:A^L--> A^L that commutes with all shift maps. A quasisturmian (QS) subshift is a shift-invariant subset obtained by mapping the trajectories of an irrational torus rotation through a partition of the torus. The image of a QS shift under a CA is again QS. We study the topological dynamical properties of CA restricted to QS shifts, and compare them to the properties of CA on the full shift A^L. We investigate injectivity, surjectivity, transitivity, expansiveness, rigidity, fixed/periodic points, and invariant measures. We also study `chopping': how iterating the CA fragments the partition generating the QS shift.Comment: 53 pages, 3 figures. To appear in Ergodic Theory and Dynamical Systems, 200

On the group of A-P diffeomorphisms and its exponential map

We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.Comment: This is a slightly amended version of the pape

Nonlinearity 8 (1995) 477491. Printed in the UK Cellular automata with almost periodic initial conditions

Abstract. Cellular automata are dynamical system on the compact metric space of subshifls. They leave many classes of sukhifts invariant. Here we show that cellul ~ automata leave 'circle subhhifls ' invariant. These are the stricUy ergodic subshifls of (0. obtained by a circle sequence zn = I,(n.e), where J is a finite union of half-open intervals. For such initial conditions, the evolution of the whole infinite confi'&tion mi be computed by evolving the finitely many parameters defining the set J. Moreover, many macroswpic quantities can be computed exactly for the infinite system. We illusmate that in one dimension by rule 18 and in WO dimensions by the Game of Life. The ides, also apply to cellular automata acting on (0,..., N- l)zd. This we illWrate by the HPP model, a lattice gas aulomalon with N = 16. AMs classification scheme numbers: 58Fxx. 70E15 1