Conservative system

Formal definition[edit]

Formally, a measurable dynamical system is conservative if and only if it is non-singular, and has no wandering sets.^[1]


A measurable dynamical system (X, Σ, μ, τ) is a Borel space (X, Σ) equipped with a sigma-finite measure μ and a transformation τ. Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ) is a measurable space. μ is a sigma-finite measure on the sigma-algebra. The space X is the phase space of the dynamical system.


A transformation (a map) ${\displaystyle \tau :X\to X}$ is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has ${\displaystyle \tau ^{-1}\sigma \in \Sigma }$. The transformation is a single "time-step" in the evolution of the dynamical system. One is interested in invertible transformations, so that the current state of the dynamical system came from a well-defined past state.


A measurable transformation ${\displaystyle \tau :X\to X}$ is called non-singular when ${\displaystyle \mu (\tau ^{-1}\sigma )=0}$ if and only if ${\displaystyle \mu (\sigma )=0}$.^[2] In this case, the system (X, Σ, μ, τ) is called a non-singular dynamical system. The condition of being non-singular is necessary for a dynamical system to be suitable for modeling (non-equilibrium) systems. That is, if a certain configuration of the system is "impossible" (i.e. ${\displaystyle \mu (\sigma )=0}$) then it must stay "impossible" (was always impossible: ${\displaystyle \mu (\tau ^{-1}\sigma )=0}$), but otherwise, the system can evolve arbitrarily. Non-singular systems preserve the negligible sets, but are not required to preserve any other class of sets. The sense of the word singular here is the same as in the definition of a singular measure in that no portion of ${\displaystyle \mu }$ is singular with respect to ${\displaystyle \mu \circ \tau ^{-1}}$ and vice versa.


A non-singular dynamical system for which ${\displaystyle \mu (\tau ^{-1}\sigma )=\mu (\sigma )}$ is called invariant, or, more commonly, a measure-preserving dynamical system.


A non-singular dynamical system is conservative if, for every set ${\displaystyle \sigma \in \Sigma }$ of positive measure and for every ${\displaystyle n\in \mathbb {N} }$, one has some integer ${\displaystyle p>n}$ such that ${\displaystyle \mu (\sigma \cap \tau ^{-p}\sigma )>0}$. Informally, this can be interpreted as saying that the current state of the system revisits or comes arbitrarily close to a prior state; see Poincaré recurrence for more.


A non-singular transformation ${\displaystyle \tau :X\to X}$ is incompressible if, whenever one has ${\displaystyle \tau ^{-1}\sigma \subset \sigma }$, then ${\displaystyle \mu (\sigma \smallsetminus \tau ^{-1}\sigma )=0}$.