Reversibility and Poincare recurrence in linear dynamical systems

In this paper, we study the Poincare recurrence phenomenon in linear dynamical systems, that is, dynamical systems whose trajectories return infinitely often to neighborhoods of their initial condition. Specifically, we provide several equivalent notions of Poincare recurrence and review sufficient conditions for nonlinear dynamical systems that ensure that the system exhibits Poincare recurrence. Furthermore, we establish necessary and sufficient conditions for Poincare recurrence in linear dynamical systems. Finally, we show that in the case of linear systems the absence of volume-preservation is equivalent to the absence of Poincare recurrence implying irreversibility of a dynamical system