Wandering set

In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is very much the opposite of a conservative system, for which the ideas of the Poincaré recurrence theorem apply. 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.

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:

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