Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time.
Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited.
Alternately, conservative systems are those to which the Poincaré recurrence theorem applies.
Commonly, such evolution is given by some differential equations, or quite often in terms of discrete time steps.
However, in the present case, instead of focusing on the time evolution of discrete points, one shifts attention to the time evolution of collections of points.
One such example would be Saturn's rings: rather than tracking the time evolution of individual grains of sand in the rings, one is instead interested in the time evolution of the density of the rings: how the density thins out, spreads, or becomes concentrated.
Over short time-scales (hundreds of thousands of years), Saturn's rings are stable, and are thus a reasonable example of a conservative system and more precisely, a measure-preserving dynamical system.
It is measure-preserving, as the number of particles in the rings does not change, and, per Newtonian orbital mechanics, the phase space is incompressible: it can be stretched or squeezed, but not shrunk (this is the content of Liouville's theorem).
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.
The transformation is a single "time-step" in the evolution of the dynamical system.
is called invariant, or, more commonly, a measure-preserving dynamical system.
A non-singular dynamical system is conservative if, for every set
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.
This is effectively the modern statement of the Poincaré recurrence theorem.
A sketch of a proof of the equivalence of these four properties is given in the article on the Hopf decomposition.
would thus contain a countably infinite union of pairwise disjoint sets that have the same
denotes the Lebesgue measure, and consider the shift operator
can be written as a countable union of wandering sets.
The Hopf decomposition states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and a wandering (dissipative) set.
A commonplace informal example of Hopf decomposition is the mixing of two liquids (some textbooks mention rum and coke): The initial state, where the two liquids are not yet mixed, can never recur again after mixing; it is part of the dissipative set.
The result, after mixing (a cuba libre, in the canonical example), is stable, and forms the conservative set; further mixing does not alter it.
In this example, the conservative set is also ergodic: if one added one more drop of liquid (say, lemon juice), it would not stay in one place, but would come to mix in everywhere.
The canonical example of an ergodic system that does not mix is the Bernoulli process: it is the set of all possible infinite sequences of coin flips (equivalently, the set
of infinite strings of zeros and ones); each individual coin flip is independent of the others.
The ergodic decomposition theorem states, roughly, that every conservative system can be split up into components, each component of which is individually ergodic.
An informal example of this would be a tub, with a divider down the middle, with liquids filling each compartment.
Clearly, this can be treated as two independent systems; leakage between the two sides, of measure zero, can be ignored.
The ergodic decomposition theorem states that all conservative systems can be split into such independent parts, and that this splitting is unique (up to differences of measure zero).
For an ergodic system, the only invariant sets are those with measure zero or with full measure (are null or are conull); that they are conservative then follows trivially from this.