Alberto A. Pinto

SMOOTHNESS OF HOLONOMIES FOR CODIMENSION 1 HYPERBOLIC DYNAMICS

Hyperbolic invariant sets {Lambda} of C1+{gamma} diffeomorphisms where either the stable or unstable leaves are 1-dimensional are considered in this paper. Under the assumption that the {Lambda} has local product structure, the authors prove that the holonomies between the 1-dimensional leaves are C1+{alpha} for some 0 < {alpha} < 1.

Fine structures of hyperbolic diffeomorphisms

HR structures.- Solenoid functions.- Self-renormalizable structures.- Rigidity.- Gibbs measures.- Measure scaling functions.- Measure solenoid functions.- Cocycle-gap pairs.- Hausdorff realizations.- Extended Livsic-Sinai eigenvalue formula.- Arc exchange systems and renormalization.- Golden tilings (in collaboration with J.P. Almeida and A. Portela).- Pseudo-Anosov diffeomorphisms in pseudo-surfaces.

Autoimmunity arising from bystander proliferation of T cells in an immune response model

We study a mathematical model of immune response by T cells where the regulatory T cells (Treg) inhibit interleukin 2 secretion. The bystander proliferation to an immune response is modelled. We consider an asymmetry reflecting that the difference between the growth and death rates can be higher for the active T cells and Tregs than for the inactive. This asymmetry leads to a better understanding of the bystander proliferation. An exposure to a pathogen results in an increased proliferation rate of the bystander T cells. If the population of the bystander T cells becomes large enough, autoimmunity can arise, eventually after a long transient period.

Unknown costs in a duopoly with differentiated products

We consider a duopoly model with unknown costs. The firms’ aims are to maximize their profits by choosing the levels of their outputs. The chooses are made simultaneously by both firms.

Bayesian price leadership

In this paper, we consider a linear price setting duopoly competition with differentiated goods and with unknown costs. The firms’ aims are to choose the prices of their products according to the well-known concept of perfect Bayesian Nash equilibrium. There is a firm (F 1) that chooses first the price p 1 of its good; the other firm (F 2) observes p 1 and then chooses the price p 2 of its good.

Teichmuller spaces and HR structures for hyperbolic surface dynamics

We construct a Teichmuller space for the C 1+ -conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structurewith each of the stable and unstablelaminations, we show that there is a one-to-one correspondence between these HR structures and the C 1+ -conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C 1+ -conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmuller space.

Rigidity of hyperbolic sets on surfaces

Given a hyperbolic invariant set of a diffeomorphism on a surface, it is proved that, if the holonomies are sufficiently smooth, then the diffeomorphism on the hyperbolic invariant set is rigid in the sense that it is C1+ conjugate to a hyperbolic affine model.

Quantity Competition in a Differentiated Duopoly

In this paper, we consider a Stackelberg duopoly competition with differentiated goods, linear and symmetric demand and with unknown costs. In our model, the two firms play a non-cooperative game with two stages: in a first stage, firm F 1 chooses the quantity, q 1, that is going to produce; in the second stage, firm F 2 observes the quantity q 1 produced by firm F 1 and chooses its own quantity q 2. Firms choose their output levels in order to maximise their profits. We suppose that each firm has two different technologies, and uses one of them following a certain probability distribution. The use of either one or the other technology affects the unitary production cost. We show that there is exactly one perfect Bayesian equilibrium for this game. We analyse the variations of the expected profits with the parameters of the model, namely with the parameters of the probability distributions, and with the parameters of the demand and differentiation.

Sensibility of the quorum growth thresholds controlling local immune responses

The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion are examined by mathematical modelling. We demonstrate that cytokine dependent growth exhibits quorum T cell population thresholds that determine whether immune responses develop on activation and whether the immune system returns to a control state. We study the effects in the quorum T cell population thresholds, by the T cell maximum growth rate, by the growth rate ratio between Tregs and T cells, by the value of the secretion rate of cytokines, and by the effectiveness of T cell secretion inhibition by Tregs.

Dynamics, Ergodic Theory, and Geometry: Solenoid functions for hyperbolic sets on surfaces

We describe a construction of a moduli space of solenoid functionsfor the C1C-conjugacy classes of hyperbolic dynamical systems f onsurfaces with hyperbolic basic sets f . We explain that if the holonomiesare sufficiently smooth then the diffeomorphism f is rigid in the sense that itis C1C conjugate to a hyperbolic affine model. We present a moduli spaceof measure solenoid functions for all Lipschitz conjugacy classes of C1C-hyperbolic dynamical systems f which have a invariant measure that is absolutelycontinuous with respect to Hausdorff measure. We extend Livˇsic andSinais eigenvalue formula for Anosov diffeomorphisms which preserve an absolutelycontinuousmeasure to hyperbolic basic sets on surfaces which possessan invariant measure absolutely continuous with respect to Hausdorff measure.