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", based on the additive topological group of p-adic integers endowed with structure of a dynamical system for the transformation, where. 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.

Construction

Nonsingular odometer

We introduce first the basic example of dyadic integers. Let be the dyadic integers, endowed with the addition which is defined for each coordinate by
where and
inductively. Let be the Borel sigma-algebra generated by the Cylinder set, with the measure and, for some independently on. Let be the transformation, where. In other words, is the transformation.
One can show that the where. Hence is -nonsingular. Moreover, one can show that has the following property: For every and natural number, the orbit of under along the steps is exactly the set . As a result, is -ergodic. Finally, is conservative since every invertible ergodic nonsingular transformation in nonatomic space is conservative.
The same construction enables to define such a system for every p-adic integers, and more generally for every product space of the form for where, endowed with the product Borel sigma-algebra and some nonatomic measure where is some measure on. The corresponding map is defined by where is the smallest index for which.

Markov odometer

Let be an ordered Bratteli–Vershik diagram, consists on a set of vertices of the form where is a singleton and on a set of edges .
The diagram includes source surjection-mappings and range surjection-mappings. We assume that are comparable if and only if.
For such diagram we look at the product space equipped with the product topology. Define "Bratteli–Vershik compactum" to be the subspace of infinite paths,
Assume there exists only one infinite path for which each is maximal and similarly one infinite path. Define the "Bratteli-Vershik map" by and, for any define, where is the first index for which is not maximal and accordingly let be the unique path for which are all maximal and is the successor of. Then is homeomorphism of.
Let be a sequence of stochastic matrices such that if and only if. Define "Markov measure" on the cylinders of by. Then the system is called a "Markov odometer".
One can show that the nonsingular odometer is a Markov odometer where all the are singeltons.