Abstract:
This talk is dedicated to the characterization of the density function of the first Poincaré returns in terms of unstable periodic orbits. We present a conjecture on how periodic orbits may be used to compute the density of the first Poincaré returns and we present numerical results that support the conjecture for some well known dynamical systems. We prove, in the case of Markov transformation under some conditions, that the density function of the first Poincaré returns is completely determined by the unstable periodic points for an element or for a perfect union of elements of the Markov partition of the map. We also discuss the extension to a more general subset S of the phase space. The close relation between periodic orbits and the Poincaré returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov-Sinai entropy. Since return times can be trivially observed and measured, this work has also application to the treatment of experimental systems. |