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.^[1]

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.