A common, discrete-time definition of wandering sets starts with a map f:X\to X of a
topological space X. A point x\in X is said to be a
wandering point if there is a
neighbourhood U of
x and a positive integer
N such that for all n>N, the
iterated map is non-intersecting: :f^n(U) \cap U = \varnothing. A handier definition requires only that the intersection have
measure zero. To be precise, the definition requires that
X be a
measure space, i.e. part of a triple (X,\Sigma,\mu) of
Borel sets \Sigma and a measure \mu such that :\mu\left(f^n(U) \cap U \right) = 0, for all n>N. Similarly, a continuous-time system will have a map \varphi_t:X\to X defining the time evolution or
flow of the system, with the time-evolution operator \varphi being a one-parameter continuous
abelian group action on
X: :\varphi_{t+s} = \varphi_t \circ \varphi_s. In such a case, a wandering point x\in X will have a neighbourhood
U of
x and a time
T such that for all times t>T, the time-evolved map is of measure zero: :\mu\left(\varphi_t(U) \cap U \right) = 0. These simpler definitions may be fully generalized to the
group action of a
topological group. Let \Omega=(X,\Sigma,\mu) be a measure space, that is, a
set with a
measure defined on its
Borel subsets. Let \Gamma be a group acting on that set. Given a point x \in \Omega, the set :\{\gamma \cdot x : \gamma \in \Gamma\} is called the
trajectory or
orbit of the point
x. An element x \in \Omega is called a
wandering point if there exists a neighborhood
U of
x and a neighborhood
V of the identity in \Gamma such that :\mu\left(\gamma \cdot U \cap U\right)=0 for all \gamma \in \Gamma-V. ==Non-wandering points==