We consider the one-dimensional Schrodinger operator with potential given by Period Doubling and Thue-Morse sequences. We describe the results of Bellisard et al. in which the spectrum of these operators are obtained through the trace map associated to this operator. We see that the spectrum for non-zero values of the potential is a Cantor set of zero Lebesque measure, and give explicit description of the spectral gaps.

We consider the classical 1D Ising model, where the coupling constants and the external magnetic field take one of two possible values at each site, according to a substitution rule. We shall introduce (briefly) the idea of a trace map corresponding to the given substitution rule and how its dynamical properties can be used to investigate the partition function and, consequently, the free energy function of the given Ising model.

Dynamical entropies are measures of the complexity of orbit structures. The topological entropy considers all the orbits, whereas the measure theoretic entropy focuses on those ``relevant" to a given invariant probability measure. The variational principle says that the topological entropy of a continuous self-map of a compact metrizable space is the supremum of the measure theoretic entropy over the set of invariant probability measures for the map.

A well known fact is that every transitive hyperbolic (Anosov) diffeomorphism has a unique invariant probability measure whose entropy equals the topological entropy. We analyze a class of deformations of Anosov diffeomorphisms containing many of the known nonhyperbolic robustly transitive diffeomorphisms. We show that these $C0$-small, but $C1$-macroscopic, deformations leave all the high entropy dynamics of the Anosov system unchanged, and that there is a partial conjugacy identifying all invariant probability measures with entropy close to the maximum for the deformation with those of the original Anosov system.

Additionally, we show that these results apply to a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana and a class originally described by Mane. In fact these methods apply to several classes of systems which are similarly derived from Anosov, i.e., produced by an isotopy from an Anosov system.

Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and Angeli) is their decomposition in terms of so-called monotone systems, which can be thought of as systems with exclusively positive feedback. In this talk I discuss some general properties of monotone dynamical systems, especially classical and recent results regarding their generic convergence towards an equilibrium. Then I will discuss the use of monotone systems to model biochemical behaviors such as global attractivity to an equilibrium, switches and oscillations under time delays.