Let A be a finite set and let ϕ:A^ℤ→ℝ be a locally constant potential. For each β>0 (‘inverse temperature’), there is a unique Gibbs measure μ_βϕ. We prove that as β→+∞, the family (μ_βϕ)_β>0 converges (in the weak-^* topology) to a measure that we characterize. This measure is concentrated on a certain subshift of finite type, which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron–Frobenius theorem for matrices à la Birkhoff. The crucial idea we bring is a ‘renormalization’ procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.

Motivated by entropy estimation from chaotic time series, we pro- vide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a n-cylinder fluctuates in the same way as the inverse measure of this n-cylinder at 'small scales', but in a different way at 'large scales'. In particular, the Rényi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more di±cult to handle theoretically. The results we obtain for return times, though less precise than for hitting times, complete the available ones.

Starting from the full shift on a finite alphabet A, by mingling some symbols of A, we obtain a new full shift on a smaller alphabet B. This amalgamation defines a factor map from (A ℕ,TA ) to (B ℕ, TB ), where TA and TB are the respective shift maps. According to the thermodynamic formalism, to each regular function (“potential”) ψ:A ℕ → ℝ, we can associate a unique Gibbs measure µψ. In this article, we prove that, for a large class of potentials, the pushforward measure µψ ∘ π−1 is still Gibbsian for a potential φ:B ℕ→ℝ having a “bit less” regularity than ψ. In the special case where ψ is a “two-symbol” potential, the Gibbs measure µψ is nothing but a Markov measure and the amalgamation π defines a hidden Markov chain. In this particular case, our theorem can be recast by saying that a hidden Markov chain is a Gibbs measure (for a Hölder potential). Introduction From different viewpoints and under different names, the so-called hidden Markov measures have received a lot of attention in the last fifty years [3]. One considers a (stationary) Markov chain (Xn ) n∈ℕ with finite state space A and looks at its “instantaneous” image Yn ≔ π(Xn ), where the map π is an amalgamation of the elements of A yielding a smaller state space, say B. It is well known that in general the resulting chain, (Yn ) n∈ℕ, has infinite memory.

Genetic regulatory networks are usually modeled by systems of coupled differential equations and by finite state models, better known as logical networks, are also used. In this paper we consider a class of models of regulatory networks which present both discrete and continuous aspects. Our models 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, according to a rule which depends on the state of the neighboring units. As a first approximation to the complete description of the dynamics of this networks we focus on a global characteristic, the dynamical complexity, related to the proliferation of distinguishable temporal behaviors. In this work we give explicit conditions under which explicit relations between the topological structure of the regulatory network, and the growth rate of the dynamical complexity can be established. We illustrate our results by means of some biologically motivated examples.

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.

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.