Definition:Factor of Measure-Preserving Dynamical System

Definition

Let $\struct {X, \BB, \mu, T}$ and $\struct {Y, \CC, \nu, S}$ be measure-preserving dynamical systems.

Then $\struct {Y, \CC, \nu, S}$ is called a factor of $\struct {X, \BB, \mu, T}$ if and only if:

there exist $X' \in \BB$, $Y' \in \CC$ and a mapping $\phi : X' \to Y'$ such that:

$\map \mu {X'} =1$

$\map \mu {Y'} = 1$

$T \sqbrk {X'} \subseteq X'$

$T \sqbrk {Y'} \subseteq Y'$

$\phi$ is measure-preserving with respect to $\struct {X', \BB_{X'}, \mu \restriction {\BB_{X'} } }$ and $\struct {Y', \CC_{Y'}, \nu \restriction {\CC_{Y'} } }$

$\forall x \in X' : \map {\phi \circ T} x = \map {S \circ \phi} x$

where

$T \sqbrk {X'}$ denotes the image of $X'$ under $T$

$S \sqbrk {Y'}$ denotes the image of $Y'$ under $S$

$\BB_{X'}$ denotes the trace $\sigma$-algebra of $X'$ in $\BB$

$\CC_{Y'}$ denotes the trace $\sigma$-algebra of $Y'$ in $\CC$

$\mu \restriction {\BB_{X'} }$ denotes the restriction of $\mu$ to $\BB_{X'}$

$\nu \restriction {\CC_{Y'} }$ denotes the restriction of $\nu$ to $\BB_{Y'}$

Also defined as

Ignoring null sets, we simply say $\phi : X \to Y$ is a measure-preserving mapping such that the diagram commutes:

$\begin{xy}\xymatrix@+1em@L+3px{

X \ar[r]^T \ar[d]_\phi & X \ar[d]^\phi \\ Y \ar[r]_S & Y }\end{xy}$

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.1$: Measure-Preserving Transformations