Measure-preserving Dynamical System

$(X,\; \backslash mathcal\{B\},\; T,\; m)$

$X$ is a set,

$\backslash mathcal\{B\}$ is a $\backslash sigma$-algebra over $X$,

$m:\backslash mathcal\{B\}\backslash rightarrow0,1$ is a probability measure, so that $m(X)=1$, and

$T:X\backslash rightarrow\; X$ is a measurable transformation which preserves the measure $m$, i. e. each measurable $A\backslash subseteq\; X$ satisfies

$m(T^\{-1\}A)=m(A).$

$m(T(A))=m(A)$

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