Basin of attraction of cycles of discretizations of dynamical systems with SRB invariant measures

Computer simulations of dynamical systems arediscretizations, where the finite space of machine arithmetic replaces continuum state spaces. So any trajectory of a discretized dynamical system is eventually periodic. Consequently, the dynamics of such computations are essentially determined by the cycles of the discretized map. This paper examines the statistical properties of the event that two trajectories generate the same cycle. Under the assumption that the original system has a Sinai-Ruelle-Bowen invariant measure, the statistics of the computed mapping are shown to be very close to those generated by a class of random graphs. Theoretical properties of this model successfully predict the outcome of computational experiments with the implemented dynamical systems.