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
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.
[3] The simplest is illustrated with the Bernoulli process.
This is the set of all infinite strings in two symbols, here denoted by
This definition extends naturally to a more general odometer defined on the product space for some sequence of integers
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.
[3] 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 product topology extends to a Borel sigma-algebra; let
The Bernoulli process is conventionally endowed with a collection of measures, the Bernoulli measures, given by
is viewed as a compact Abelian group.
is also the base space for the dyadic integers; however, the dyadic integers are endowed with a metric, the p-adic metric, which induces a metric topology distinct from the product topology used here.
Increment-by-one is then called the (dyadic) odometer.
It is called the odometer due to how it looks when it "rolls over":
The same construction enables to define such a system for every product of discrete spaces.
A special case of this is the Ornstein odometer, which is defined on the space with the measure a product of A concept closely related to the conservative odometer is that of the abelian sandpile model.
This model replaces the directed linear sequence of finite groups constructed above by an undirected graph
Transition functions are defined by the graph Laplacian.
Next, the sandpile models in general use undirected edges, so that the wrapping of the odometer redistributes in all directions.
A third difference is that sandpile models are usually not taken on an infinite graph, and that rather, there is one special vertex singled out, the "sink", which absorbs all increments and never wraps.
The sink is equivalent to cutting away the infinite parts of an infinite graph, and replacing them by the sink; alternately, as ignoring all changes past that termination point.
be an ordered Bratteli–Vershik diagram, consists on a set of vertices of the form
The diagram includes source surjection-mappings
Define "Bratteli–Vershik compactum" to be the subspace of infinite paths, Assume there exists only one infinite path
is maximal and similarly one infinite path
Define "Markov measure" on the cylinders of
