Edgardo Ugalde

Projection of Markov Measures May Be Gibbsian

We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Hölder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.

Dynamical complexity of discrete-time regulatory networks

Genetic regulatory networks are usually modelled by systems of coupled differential equations, and more particularly by systems of piecewise affine differential equations. Finite state models, better known as logical networks, are also used. In this paper we present a class of models of regulatory networks which may be situated in the middle of the spectrum; they present both discrete and continuous aspects. They consist of a network of units, whose states are quantified by a continuous real variable. The state of each unit in the network evolves according to a contractive transformation chosen from a finite collection of possible transformations. The particular transformation chosen at each time step depends on the state of the neighbouring units. In this way we obtain a network of coupled contractions. As a first approximation to the complete description of the dynamics we shall focus on a global invariant: the dynamical complexity of the system. This is a well-studied notion in the framework of the theory of dynamical systems, and it is related to the proliferation of distinguishable temporal behaviour. The main motivation of this work is to find explicit relations between the topological structure of the regulatory network and the growth rate of the dynamical complexity. In this paper we derive general upper bounds for the dynamical complexity for networks of arbitrary size, and we exhibit specific instances of constraints imposed on the complexity growth by the structure of the underlying network.

A cryptosystem based on cellular automata

Cryptosystems for binary information are based on two primitives: an indexed family of permutations of binary words and a generator of pseudorandom sequences of indices. A very efficient implementation of the primitives is constructed using the phenomenon of synchronization in cellular automata. (c) 1998 American Institute of Physics.

Fractal Dimensions for Poincaré Recurrences

1. Introduction Part 1: Fundamentals 2. Symbolic Systems 3. Geometric Constructions 4. Spectrum of Dimensions for Recurrences Part II: Zero-Dimensional Invariant Sets 5. Uniformly Hyperbolic Repellers 6. Non-Uniformly Hyperbolic Repellers 7. The Spectrum for a Sticky Set 8. Rhythmical Dynamics Part III: One-Dimensional Systems 9. Markov Maps of the Interval 10. Suspended Flows Part IV: Measure Theoretical Results 11. Invariant Measures 12. Dimensional for Measures 13. The Variational Principle Part V: Physical Interpretation and Applications 14. Intuitive Explanation 15. Hamiltonian Systems 16. Chaos Synchronization Part VI: Appendices 17. Some Known Facts About Recurrences 18. Birkhoffs Individual Theorem 19. The SMB Theorem 20. Amalgamation and Fragmentation Index

Synchronization of cellular automaton pairs

The phenomenon of synchronization in pairs of cellular automata coupled in a driver-replica mode is studied. Necessary and sufficient conditions for synchronization in linear cellular automaton pairs are given. The couplings that make a pair synchronize are determined for all linear elementary cellular automata. (c) 1998 American Institute of Physics.

A Spatially Extended Model for Residential Segregation

We analyze urban spatial segregation phenomenon in terms of the income distribution over a population, and an inflationary parameter weighting the evolution of housing prices. For this, we develop a discrete spatially extended model based on a multiagent approach. In our model, the mobility of socioeconomic agents is driven only by the housing prices. Agents exchange location in order to fit their status to the cost of their housing. On the other hand, the price of a particular house depends on the status of its tenant, and on the neighborhood mean lodging cost weighted by a control parameter. The agents dynamics converges to a spatially organized configuration, whose regularity is measured by using an entropy-like indicator. This simple model provides a dynamical process organizing the virtual city, in a way that the population inequality and the inflationary parameter determine the degree of residential segregation in the final stage of the process, in agreement with the segregation-inequality thesis put forward by Douglas Massey.

On the asymptotic properties of piecewise contracting maps

ABSTRACT We are interested in the phenomenology of the asymptotic dynamics of piecewise contracting maps. We consider a wide class of such maps and we give sufficient conditions to ensure some general basic properties, such as the periodicity, the total disconnectedness or the zero Lebesgue measure of the attractor. These conditions show in particular that a non-periodic attractor necessarily contains discontinuities of the map. Under this hypothesis, we obtain numerous examples of attractors, ranging from finite to connected and chaotic, contrasting with the (quasi-)periodic asymptotic behaviours observed so far.

On the density of directional entropy in lattice dynamical systems

We study the direction dependence of the density of directional entropy in lattice dynamical systems. We show that if the dynamics is homogeneous and continuous, then this density does not depend on the direction in space–time. By using symbolic dynamics we derive formulae for the density for weakly coupled hyperbolic maps. As a corollary, we present examples where this density actually depends on the direction, provided that individual subsystems are sufficiently different.

Traveling patterns in cellular automata

A method to identify the invariant subsets of bi-infinite configurations of cellular automata that propagate rigidly with a constant velocity nu is described. Causal traveling configurations, propagating at speeds not greater than the automaton range, mid R:numid R:</=r, are considered. The sets of traveling configurations are presented by finite automata and its topological entropy is calculated. When the invariant subset of traveling configurations has nonzero topological entropy, the dynamics is dominated by the interaction of domains, composed of traveling patterns of finite size. The sets of traveling patterns and domains are presented by finite automata. End-resolving CA are shown to always have sets of traveling configurations that are spatially periodic with zero entropy, except possibly for traveling configurations at top speed. The elementary CA are examined exhaustively along these lines. (c) 1996 American Institute of Physics.

Self-similarity and finite-time intermittent effects in turbulent sequences

We discuss two questions related to the finite-time behaviour of a sequence that generates a self-similar system. The first one concerns the way to approach the self-similarity exponent associated with the system, from the analysis of the sequences observed during a finite time. After this approximation procedure we established a criterion to decide whether the analysed sequence can be considered to be a finite-time subsequence of a generating sequence for a self-similar system. In the second part of this work, assuming the asymptotic self-similarity for the system generated by a sequence, we study the conditions ensuring the appearance of anomalous scaling on the structure functions, due to finite-time effects. We use this method to show that, in the case of a real turbulent sequence, anomalous scaling is not incompatible with asymptotic self-similarity.