In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposite of a conservative system, to which the Poincaré recurrence theorem applies. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927.^{[citation needed]}

A common, discrete-time definition of wandering sets starts with a map ${\displaystyle f:X\to X}$ of a topological space X. A point ${\displaystyle x\in X}$ is said to be a wandering point if there is a neighbourhood U of x and a positive integer N such that for all ${\displaystyle n>N}$, the iterated map is non-intersecting:

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 ${\displaystyle (X,\Sigma ,\mu )}$ of Borel sets ${\displaystyle \Sigma }$ and a measure ${\displaystyle \mu }$ such that

for all ${\displaystyle n>N}$. Similarly, a continuous-time system will have a map ${\displaystyle \varphi _{t}:X\to X}$ defining the time evolution or flow of the system, with the time-evolution operator ${\displaystyle \varphi }$ being a one-parameter continuous abelian group action on X:

In such a case, a wandering point ${\displaystyle x\in X}$ will have a neighbourhood U of x and a time T such that for all times ${\displaystyle t>T}$, the time-evolved map is of measure zero:

These simpler definitions may be fully generalized to the group action of a topological group. Let ${\displaystyle \Omega =(X,\Sigma ,\mu )}$ be a measure space, that is, a set with a measure defined on its Borel subsets. Let ${\displaystyle \Gamma }$ be a group acting on that set. Given a point ${\displaystyle x\in \Omega }$, the set

is called the trajectory or orbit of the point x.

An element ${\displaystyle x\in \Omega }$ is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in ${\displaystyle \Gamma }$ such that