Dynamical Systems

The following sections are included:

• FOREWORD

• List of Participants

• CONTENTS

An important part of the work of Gérard Rauzy concerns the study of statistical properties of arithmetical sequences, with a special interest for those obtained by some classes of simple algorithmic constructions.

I want to mention three such problems studied by Rauzy. The first one concerns the combinatorics of sequences of integers generated by some greedy algorithms, the second the geometric representation of sequences generated by substitutions on finite alphabets and the last asks about the repartition modulo 1 of sequences generated by denumerable automata…

Synchronization is shown to occur in spatially extended systems under the effect of additive spatio-temporal noise. In analogy to low dimensional systems, synchronized states are observable only if the maximum Lyapunov exponent Λ is negative. However, a sufficiently high noise level can lead, in map with finite domain of definition, to nonlinear propagation of information, even in non chaotic systems. In this latter case the transition to synchronization is ruled by a new ingredient : the propagation velocity of information V_F. As a general statement, we can affirm that if V_F is finite the time needed to achieve a synchronized trajectory grows exponentially with the system size L, while it increases logarithmically with L when, for sufficiently large noise amplitude, V_F = 0.

In dynamical systems with many degrees of freedom the Lyapunov exponent is not always able to characterize, even in a qualitative way, the main spatio-temporal or macroscopic behavior. We study two examples which show the inadequacy of the standard Lyapunov analysis. We introduce a ‘spatial’ Lyapunov exponent to characterize the spatial complex behavior of non-chaotic but convectively unstable flow systems. This quantity gives a relation between the sensitivity on the boundary conditions and the co-moving Lyapunov exponent. We also study systems of coupled maps in which some global quantities with typical times much longer than the inverse of the Lyapunov exponent appear. A generalization of the Lyapunov exponent for finite perturbations can characterize, in a consistent way, such a macroscopic behavior, even in the absence of explicit equations for the time evolution of these global observables.

We apply the notions of ε-entropy per unit volume of Shannon and Kolmogorov to the case of attractors of non linear parabolic evolutions in unbounded domains. We extend these notions to the topological entropy and show how it can be measured using a discrete sampling.

We discuss mathematical issues suggested by the processes of first-language acquisition and language change. We present a model of language acquisition with two components: A probability measure describing sentence selection by a native speaker, and an identification principle modeling how the child chooses an element of the finite set of natural grammars. More generally, we present an approach to the problem of classifying existing evidence to choose among a finite set of policies in the presence of possibly conflicting hints.

A novel approach to quantum tight-binding models is discussed. It is based on a representation of stationary Schrödinger equation in terms of two-dimensional Hamiltonian maps. This reduction allows one to study localization properties of eigenstates by exploring trajectories of classical maps in the phase space. Application of this approach to disordered potentials as well as to potentials with correlated disorder is discussed. Particular interest is the existence of mobility edges in 1D geometry for potentials with long-range correlations. It was shown how to construct such potentials practically.

The concatenation method for constructing generic points of an arbitrary stationary probability measure is discussed within the framework of Large Deviation Theory. We consider in details the case of k-Markov measures. We compute the Hausdorff dimension of the set of generic points in the general case. The emphasis is put on the main features of this construction and on the connection with basic ideas in Information Theory around the Ergodic Theorem of Shannon-McMillan-Breiman.

Radu Orendovici died in a tragic accident in mountains near Marseille-Luminy where he was invited to give a talk at the International conference “From Crystal to Chaos”. He was a bright very promising graduate student who proved a number of prolific results in the theory of dynamical systems. Radu was a remarkable personality, gentle, a bit shy, but very kind and very responsible. He is survived by his wife and a child who came from Romania…

Chaotic neuron models can display a wide variety of coherent spatio-temporal patterns in networks with large number of units. We present a study of pattern formation and regularization phenomena using chaotic Hindmarsh-Rose model neurons. While the parameters of each neuron were initially set in the range of values where individual chaotic spiking-bursting activity was observed, the colective activity inside the networks produced coherent and well defined spatio-temporal patterns with the activity of each unit regularized. We investigated several types of network architectures and connections and we report an explanation for the observed phenomena.

We consider the micro-canonical ensemble of classical Hamiltonian mechanics from a geometric and dynamical point of view. We show how various thermodynamic quantities may be calculated within the micro-canonical ensemble itself, without making explicit reference to the canonical ensemble. We rederive formulas by Lebowitz et al⁵ and Pearson et al⁶, relating e.g. specific heat to fluctuations in the kinetic energy.

In neuroscience, optics and condensed matter there is ample physical evidence for multistable dynamical systems, that is, systems with a large number of attractors. The known mathematical mechanisms that lead to multiple attractors are homoclinic tangencies and stabilization, by small perturbations or by coupling, of systems possessing a large number of unstable invariant sets. A short review of the existent results is presented, as well as two new results concerning the existence of a large number of stable periodic orbits in a perturbed marginally stable dissipative map and an infinite number of such orbits in two coupled quadratic maps working on the Feigenbaum accumulation point.

Examples are presented of minimal subshifts with positive entropy, and their Afraimovich-Pesin capacities are computed. It is shown that the lower capacity can be strictly smaller than the entropy.

Discrete sets of numbers related to some irrational number β, called the β-integers ℤ_β, are a natural tool to describe quasicrystals. We prove that, when β is a quadratic Pisot unit, the set ℤ_β is a group isomorphic to the set of integers ℤ.

Let and F be finite sets and suppose for each we have a map φ_a : F → F. Suppose ω = ω₁ω₂ω₃ … is a sequence in which is ultimately primitive substitutive, i.e., a tail of ω is the image (under a letter to letter morphism) of a fixed point of a primitive substitution. We show that the induced sequence of iterates is also ultimately primitive substitutive, and is itself primitive substitutive if it is recurrent.

Using Modified Jacobi-Perron algorithm T on [0, 1]² a real number x ∈ [0,1] can be expanded arithmetically where θ_i := max (α_i, β_i), (α_i, β_i) := Tⁱ (α, β). On the assumption that (α, β) is a purely periodic point with respect to T, we know that λ = θθ₁ …θ_k−1 ( where k is the period ) is a cubic number and know also that the number x belongs to the cubic field Q(λ) if the sequence is periodic. In this paper we claim on the above assumption that (1) the number x ∈ Q(λ) ∩ [0, 1] if and only if the sequence periodic. Moreover, (2) the number x ∈ Q(λ) ∩ [0, 1] and x is reduced if and only if the sequence is purely periodic.

We show that for a large class of minimal dynamical systems, the AP dimension of recurrence as defined in ⁶ is zero. For subshifts, we consider a variant of this dimension based on cylinder partitions. The dimension obtained is topological invariant less or equal to topological entropy. We show an example of Toeplitz subshifts for which it is positive and another example where it is zero, while the topological entropy is positive.

The phenomenon of synchronization is studied in pairs of elementary cellular automata (CA) that are coupled in a drive/response mode. Sufficient and necessary conditions are given for the synchronization of linear elementary CA. By using synchronization, two large families of indexed permutations of binary words and a random number generator are assembled in the form of a single CA array as primitives for block cryptosystems. Numerical results show excellent statistical behavior of the random number generator. The overall architecture is very well suited for hardware implementations.

We discuss the one-dimensional cellular automaton

We study irrational numbers whose continued fraction expansion is an automatic sequence on a finite alphabet. By using a theorem due to W.Schmidt, we prove the transcendence of a big class of them.

We consider certain partition of a lattice into c (2 < c ≤ ∞) parts together with its characteristic word W(A; c) on the lattice, which are determined uniquely by c and a given square matrix A of size s × s with integer entries belonging to a class (Bdd). We define automata, and substitutions of dimension s together with their conjugates. We show that the word W(A; c) becomes the fixed point of a substitution for A belonging to a subclass of (Bdd). The hermitian canonical forms of integer matrices play an important role in some cases for finding substitutions. We give a theorem which discloses a p-adic link with hermitian canonical forms. We give two definitions for periodicity: Ψ-periodicity, and Σ-periodicity, respectively, for words, and for tilings of higher dimension; the non-Σ-periodicity strongly requires non-periodicity, so that it excludes some non-periodic words in the usual sense. We also consider certain Voronoi tessellations coming from a word W(A; c). We show that some of the words W(A; c), and the tessellations are non-Σ-periodic.

The following sections are included:

• Introduction

• Definitions and notation
  • Normal numbers
  • Paths on multidigraphs
  • The de-Bruijn multidigraphs
  • The Large Deviation Property

• Statement of the results

• Computations

• Concluding Remarks

• References

The following is an outline of results of our research on a class of nonergodic piecewise affine maps of the torus, the proofs of which will appear elsewhere. These maps exhibit highly complex and little understood behavior. They have appeared in the engineering literature (e.g., see ¹²) and the physics literature (e.g., see ¹ ). We present computer graphics of some examples and the results concerning the one we understand best.

We describe an attracting piecewise rotation with two atoms with a self-similar structure of periodic domains on its attractor which resembles Sierpinski's gasket. Besides its natural beauty, this example appears a return map in certain piecewise affine maps on the torus which have been studied by Adler, Kitchens, and Tresser, and by a number of other researchers advancing the theory of digital filters.

The well known relation between billiard orbits in a rational polygon and the geodesies on a translation surface justifies a study of closed geodesies on flat surfaces. In this contribution we consider the situation when a translation surface covers another one. We investigate relations between closed geodesies in the two surfaces. In particular, we obtain inequalities relating the growth rates of the number of closed geodesies in a surface and its covering. We apply these results to polygonal billiards.

We prove that ergodicity of the horocycle flow on a surface of constant negative curvature is equivalent to ergodicity of the associated boundary action. As a corollary we obtain ergodicity of the horocycle flow on several large classes of covering surfaces.

Using a classical notion of means we introduce a family of dynamical systems. We analyze the simplest two-dimensional case.

Interval translation mappings are not necessarily invertible local isometries of the interval which where recently introduced by Boshernitzan and Kornfeld ¹. Our main result is a sharp upper bound on the number of minimal sets such a mapping can have.

Following is the abstract of my conference talk given at the meeting “Dynamical systems: From Crystal to chaos” organized at the University of Marseille, Luminy, France in July, 1998. The results below are joint achievements with Domokos Szász, Mathematical Institute of the Hungarian Academy of Sciences.