**Asymptotic theory with statistical applications**

We have proved, under certain regularity conditions and convenient decay rates of the covariances, the almost sure consistency of the kernel estimator of the p-dimensional marginal distribution function of a stationary associated sequence. Asymptotic normality for the finite dimensional distributions results has been established. We characterized also the asymptotic behavior of the MSE and has been shown, under some stronger assumptions on the covariance decay rate, that is of the same order as for independent sequences. Some procedures involving simulation has been applied to compare empirically kernel estimation with other classes of estimators for distributions functions.

We have been also studying the extremal behavior of models involving dependent sequences and we are proceeding the study of Bayes Factors in statistical inference.

In the year of 2006, research aims of some people in this task are in the field of Biostatistics and Bioinformatics.

In fact our aim is now on applications of statistics to life sciences in general.

Therefore we began to study this kind of applications, and we are involved with people from Spain

studying statistical applications to Medicine, Economics, Finance, Engineering and Environmental Sciences.

**Distributions for the horocycle flow**

We investigated the Holder regularity of invariant distributions for the horocycle flow on compact hyperbolic surfaces (S.Cosentino, "Nonlinearity" 18 (2005), 2715-2726), showing that those invariant distributions, as described by L. Flaminio and G. Forni (L.Flaminio and G.Forni, "Duke Math. J." 119 (2003), 465-526), can be also obtained from certain distributions that generalize Patterson-Sullivan's conformal measures on the ideal boundary of the hyperbolic plane.

**Topological Markov chains**

We introduce and study, from a combinatorial-topological viewpoint, some semigroups of continuous non-deterministic dynamical systems. Combinatorial stability, i.e. the attractors' combinatorics persistence, is characterized and its genericity established. Some implications on topological (deterministic) dynamics are drawn.

**Branching processes between maximum and extinction**

We use multi-type branching processes to model the evolution of a population that starts only with subcritical individuals, and therefore dies with probability one, but mutations occuring during the reproduction process may lead to supercritical individuals that are able to start a population that can escape extinction. Assuming small mutation probabilities, we obtained approximations for the probability of extinction/escape of such populations and for distribution of the time to produce a successful mutant, i.e, the first supercritical individual that did not get extinct. We also have simulation results on the time it takes for the number of individuals in a supercritical single-type process to cross certain high levels and, these combined with the previous results, allowed us to obtain results for a two-type decomposable branching process.