Resumen:
This work investigates how the fractal geometry of a space influences the speed at which orbits return to small neighborhoods of their initial points. The object of study is a dynamical system that preserves a probability measure. The classical Poincaré Recurrence Theorem is a qualitative result, guaranteeing only that, for almost every point in the system, the return time is finite. Here, we move toward a quantitative description: we show that, along a subsequence, the distance between a typical orbit at time n and its initial point decays in a polynomial or subpolynomial manner, with the decay rate estimated by the Hausdorff dimension of the space. More precise estimates can be obtained by considering the Hausdorff dimension of the invariant measure. In addition, we use the theory developed for the deterministic case to study the random setting, in which orbits are generated by independent and identically distributed choices of functions. We show that the recurrence of random orbits exhibits the same behavior observed in the deterministic case.
