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 , where . 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. Several kinds of non-singular odometers may be defined. These are sometimes referred to as adding machines. The simplest is illustrated with the Bernoulli process. This is the set of all infinite strings in two symbols, here denoted by endowed with the product topology. This definition extends naturally to a more general odometer defined on the product space for some sequence of integers with each The odometer for for all is termed the dyadic odometer, the von Neumann–Kakutani adding machine or the dyadic adding machine. The topological entropy of every adding machine is zero. Any continuous map of an interval with a topological entropy of zero is topologically conjugate to an adding machine, when restricted to its action on the topologically invariant transitive set, with periodic orbits removed. The set of all infinite strings in strings in two symbols has a natural topology, the product topology, generated by the cylinder sets. The product topology extends to a Borel sigma-algebra; let denote that algebra. Individual points are denoted as The Bernoulli process is conventionally endowed with a collection of measures, the Bernnoulli measures, given by and , for some independent of .