A new proof of the Herman-Avila-Bochi formula for Lyapunov exponents of SL(2,R)-cocycles

We study the geometry of the action of SL(2,R) on the projective line in order to present a new and simpler proof of the Herman-Avila-Bochi formula. This formula gives the average Lyapunov exponent of a class of 1-families of SL(2,R)-cocycles.Comment: 13 pages, 2 figure

Emerging collective behavior in a simple artificial financial market

We consider a simple model of a society of economic agents, where each can invest a discrete quantity. Interactions among agents happen in a neighborhood and depend on the motivation level (insider information, economy prospects, etc.). The profit of the group fluctuates stochastically and is used to update individual motivations. We analyze the behavior, as a function of time, of the global persistence, given the initial quantity of money invested. Our simulations show that this quantity – a measure of the probability that the amount of money of the entire group remains at least equal to the initial amount – has a power law updating behavior. We have also performed simulations with heterogeneous agents, including deceiver and conservative agents. We show that, although there is no regular pattern regarding the average wealth, robust power laws for persistence exist, indicating that this can be used to model the emerging collective behavior. Besides, the updating of motivation and the presence of conservatives and deceivers is remarkable and has an influence on the persistence