For a Topology of Dynamical Systems

Abstract

Dynamical systems are mathematical objects meant to formally capture the dynamical features of deterministic systems. They are commonly defined as ordered pairs of the form DS = (S, (f ^t) _t ∈ T ), where S is a non-empty set of states or points called the state space, and (f ^t) _t ∈ T is a family of functions on S, indexed by T, called state transitions. For every t ∈ T, the state transition f ^t is said to have duration t, where the time set T is usually taken to be a set of numbers, such as the reals $\mathcal{R}$, the non-negative reals $\mathcal{R}^{0}$, the integers $\mathcal{Z}$, or the non-negative integers $\mathcal{Z}^{0}$.