In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium.

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

with the following structure:

One may ask why the measure preserving transformation is defined in terms of the inverse ${\displaystyle \mu (T^{-1}(A))=\mu (A)}$ instead of the forward transformation ${\displaystyle \mu (T(A))=\mu (A)}$. This can be understood intuitively.

Consider the typical measure on the unit interval ${\displaystyle [0,1]}$, and a map ${\displaystyle Tx=2x\mod 1={\begin{cases}2x{\text{ if }}x<1/2\\2x-1{\text{ if }}x>1/2\\\end{cases}}}$. This is the Bernoulli map. Now, distribute an even layer of paint on the unit interval ${\displaystyle [0,1]}$, and then map the paint forward. The paint on the ${\displaystyle [0,1/2]}$ half is spread thinly over all of ${\displaystyle [0,1]}$, and the paint on the ${\displaystyle [1/2,1]}$ half as well. The two layers of thin paint, layered together, recreates the exact same paint thickness.

More generally, the paint that would arrive at subset ${\displaystyle A\subset [0,1]}$ comes from the subset ${\displaystyle T^{-1}(A)}$. For the paint thickness to remain unchanged (measure-preserving), the mass of incoming paint should be the same: ${\displaystyle \mu (A)=\mu (T^{-1}(A))}$.

Consider a mapping ${\displaystyle {\mathcal {T}}}$ of power sets:

Consider now the special case of maps ${\displaystyle {\mathcal {T}}}$ which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends ${\displaystyle X}$ to ${\displaystyle X}$ (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map ${\displaystyle T:X\to X}$ by writing ${\displaystyle {\mathcal {T}}(A)=T^{-1}(A)}$. Of course, one could also define ${\displaystyle {\mathcal {T}}(A)=T(A)}$, but this is not enough to specify all such possible maps ${\displaystyle {\mathcal {T}}}$. That is, conservative, Borel-preserving maps ${\displaystyle {\mathcal {T}}}$ cannot, in general, be written in the form ${\displaystyle {\mathcal {T}}(A)=T(A);}$.