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. 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 f : X → X {displaystyle f:X o X} of a topological space X. A point x ∈ X {displaystyle xin 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 {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 ( X , Σ , μ ) {displaystyle (X,Sigma ,mu )} of Borel sets Σ {displaystyle Sigma } and a measure μ {displaystyle mu } such that Similarly, a continuous-time system will have a map φ t : X → X {displaystyle varphi _{t}:X o X} defining the time evolution or flow of the system, with the time-evolution operator φ {displaystyle varphi } being a one-parameter continuous abelian group action on X: In such a case, a wandering point x ∈ X {displaystyle xin X} will have a neighbourhood U of x and a time T such that for all times t > T {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 Ω = ( X , Σ , μ ) {displaystyle Omega =(X,Sigma ,mu )} be a measure space, that is, a set with a measure defined on its Borel subsets. Let Γ {displaystyle Gamma } be a group acting on that set. Given a point x ∈ Ω {displaystyle xin Omega } , the set is called the trajectory or orbit of the point x. An element x ∈ Ω {displaystyle xin Omega } is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in Γ {displaystyle Gamma } such that for all γ ∈ Γ − V {displaystyle gamma in Gamma -V} .