Markov odometer

In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer.

The basic example of such system is the "nonsingular odometer", which is an additive topological group defined on the product space of discrete spaces, induced by addition defined as ${\displaystyle x\mapsto x+{\underline {1}}}$, where ${\displaystyle {\underline {1}}:=(1,0,0,\dots )}$. This group can be endowed with the structure of a dynamical system; the result is a conservative dynamical system.

The general form, which is called "Markov odometer", can be constructed through Bratteli–Vershik diagram to define Bratteli–Vershik compactum space together with a corresponding transformation.