Common assumptions on the source producing the words inserted in a suffix trie with n leaves lead to a log n height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of n and another one whose saturation level is negligible with respect to log n. Both are built from VLMC (Variable Length Markov Chain) probabilistic sources and are easily extended to families of tries having the same properties. The first example corresponds to a "logarithmic infinite comb" and enjoys a non uniform polynomial mixing. The second one corresponds to a "factorial infinite comb" for which mixing is uniform and exponential.

In this article we study the continuity properties of trajectories for some randomly sampled series of functions, ∑ a_kf(α X_k(ω))where (a_k)_k⩾0 is a sequence of complex numbers, (X_k)_k⩾0 is a sequence of real independent random variables, $f$ is a real valued function with period one and summable Fourier coefficients. We obtain almost sure continuity results for these periodic or almost periodic series for a large class of functions f, where the "almost sure" does not depend on the function. We show optimality of the results in some cases.

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the "comb" and the "bamboo blossom", we find a necessary and sufficient condition for the existence and the uniqueness of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences.

To model the invasion of Prunus serotina invasion within a real forest landscape we built a spatially explicit, non-linear Markov chain which incorporated a stage-structured population matrix and dispersal functions. Sensitivity analyses were subsequently conducted to identify key processes controlling the spatial spread of the invader and testing the hypothesis that the landscape invasion patterns are driven by disturbance patterns, local demographical processes controlling propagule pressure, habitat suitability, and long-distance dispersal. When offspring emigration was considered as a density-dependent phenomenon, local demographic factors generated invasion patterns at larger spatial scales through three factors: adult longevity; adult fecundity; and the intensity of self-thinning during stand development. Three other factors acted at the landscape scale: habitat quality, which determined the proportion of the landscape mosaic which was potentially invasible; disturbances, which determined when suitable habitats became temporarily invasible; and the existence of long distance dispersal events, which determined how far from the existing source populations new founder populations could be created. As a flexible “all-in-one” model, PRUNUS offers perspectives for generalization to other plant invasions, and the study of interactions between key processes at multiple spatial scales.

In this paper are characterized the polynomials, in terms of their coefficients, that have all their orbits dense in the set of 3-adic integers ℤ₃

The effect of environmental heterogeneity on spatial spread of invasive species has received little attention in the literature. Altering landscape heteroge- neity may be a suitable strategy to control invaders in man-made landscapes. We use a population-based, spatially realistic matrix model to explore mechanisms underlying the observed invasion patterns of an alien tree species, Prunus serotina Ehrh., in a heterogeneous managed forest. By altering several parameters in the simulation, we test for various hypotheses regarding the role of several mechanisms on invasion dynamics, including spatial heterogeneity, seed dispersers, site of first introduction, large-scale natural disturbances, and forest management. We observe that landscape hetero- geneity makes the invasion highly directional resulting from two mechanisms: (1) irregular jumps, which occur rarely via long-distance dispersers and create new founder populations in distant suitable areas, and (2) regular, continuous diffusion toward adjacent cells via short- and mid-distance vectors. At the landscape scale, spatial heterogeneity increases the invasion speed but decreases the final invasion extent. Hence, natural disturbances (such as severe storms) appear to facilitate invasion spread, while forest management can have contrasting effects such as decreasing invasibility at the stand scale by increasing the proportion of light interception at the canopy level. The site of initial introduction influences the invasion process but without altering the final outcome. Our model represents the real landscape and incorporates the range of dispersal modes, making it a powerful tool to explore the interactions between environmental heterogeneity and invasion dynamics, as well as for managing plant invaders.

Biological invasions are widely accepted as having a major impact on ecosystem functioning worldwide, giving urgency to a better understanding of the factors that control their spread. Modelling tools have been developed for this purpose but are often discrete-space, discrete- time spatial-mechanistic models that adopt a computer simulation approach and resist mathematical analysis. We constructed a simple demographic matrix model to explore the local population dynamics of an invasive species with a complex life history and whose invasive success depends on resource availability, which occurs stochastically. As a case study we focused on the American black cherry (Prunus serotina Ehrh.), a gap-dependent tree able both to constitute a long-living seedling bank under unfavourable light conditions and to resprout vigorously once cut-down, which is invading European temperate forests. The model used was a stage-classified matrix population model (i.e., Lefkovitch matrix), inte- grating environmental stochasticity. Stochastic matrix projection analysis was combined with elasticity analysis and stochastic simulations to search for the species "Achille heel". As expected, the population growth rate (i.e., Lyapunov exponent), which measures the risk of P. serotina invasion at the stand scale, increased with light frequency. There was a crit- ical value above which the population of P. serotina explodes and below which it locally goes extinct. The resprouting capacity usually speed up the invasion but appeared to play a minor role. The mean duration of stand invasion was measured and important life stage transitions that mostly contribute to the local stochastic growth rate were identified. Some relevant management implications are discussed and the interest of such models for the understanding of demographic characteristics of invasive species is stressed.

We study a class of geometric Lorenz flows, introduced independently by Aframovič, Bykov and Sil'nikov and by Guckenheimer and Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.

Given a piecewise invertible map T:X → X and a weight g:X → ]0, ∞[, a conformal measure ν is a probability measure on X such that, for all measurable A ⊂ X with T:A → TA invertible,
ν(TA)=λ ∫_A g^-1 dν
with λ>0. Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that log g has bounded distortion and an irreducibility condition, we build such a conformal measure.

For non markovian, piecewise monotonic maps of the interval associated to a potential, we prove that the law of the entrance time in a cylinder, when renormalized by the measure of the cylinder, converges to an exponential law for almost all cylinders. Thanks to this result, we prove that the fluctuations of R_n, first return time in a cylinder, are lognormal.

This thesis deals with some statistical properties of piecewise invertible, non markovian dynamical systems. The main problem for such systems is that the transfer operator does not act on the space of continuous functions. In the first part, studying the dual of the transfer operator leads to the existence of a conformal measure for general potentials. The main hypothesis on the system is : the topological pressure of the boundary is strictly less than the total pressure (this means that the discontinuities should not accumulate everywhere). The second part is devoted to the study of the transfer operator itself, acting on a space of functions with bounded variation. The study of the spectrum of the operator, together with a condition on the lack of markoviannity of the system, enable to prove the existence of an invariant measure, absolutely continuous with respect to the conformal one, and the exponential decay of correlations. In the last part, the fact that the system is α-mixing is essential to prove that the statistics of entrance times in a cylinder goes asymptotically to an exponential law. Showing that the invariant measure is not too different from a Gibbs measure, the fluctuations of return times into a cylinder around the entropy turn out to be lognormal.