Topological Dynamics: A Survey

1 Glossary Attractor An invariant subset for a dynamical system such that points sufficiently close to the set remain close and approach the set in the limit as time tends to infinity. Dynamical System A model for the motion of a system through time. The time variable is either discrete, varying over the integers, or continuous, taking real values. Our systems are deterministic, rather than stochastic, so the the future states of the system are functions of the past. Equilibrium An equilibrium, or a fixed point, is a point which remains at rest for all time. Invariant Set A subset is invariant if the orbit of each point of the set remains in the set at all times, both positive and negative. The set is + invariant, or forward invariant, if the forward orbit of each such point remains in the set. Lyapunov Function A continuous, real-valued function on the state space is a Lyapunov function when it is non-decreasing on each orbit as time moves forward. Orbit The orbit of an initial position is the set of points through which the system moves as time varies positively and negatively through all values. The forward orbit is the subset associated with positive times. Recurrence A point is recurrent if it is in its own future. Different concepts of recurrence are obtained from different interpretations of this vague description. Repellor A repellor is an attractor for the reverse system obtained by changing the sign of the time variable. Transitivity A system is transitive if every point is in the future of every other point. Periodicity, minimality, topological transitivity and chain transitivity are different concepts of transitivity obtained from different interpretations of this vague description.