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 formula_1, where formula_2. 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.
Nonsingular odometers.
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 formula_3 endowed with the product topology. This definition extends naturally to a more general odometer defined on the product space
for some sequence of integers formula_5 with each formula_6
The odometer for formula_7 for all formula_8 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.
Dyadic odometer.
The set of all infinite strings in strings in two symbols formula_3 has a natural topology, the product topology, generated by the cylinder sets. The product topology extends to a Borel sigma-algebra; let formula_10 denote that algebra. Individual points formula_11 are denoted as formula_12
The Bernoulli process is conventionally endowed with a collection of measures, the Bernnoulli measures, given by formula_13 and formula_14, for some <math>0 independent of formula_8. The value of formula_16 is rather special; it corresponds to the special case of the Haar measure, when formula_17 is viewed as a compact Abelian group. Note that the Bernoulli measure is "not" the same as the 2-adic measure on the dyadic integers! Formally, one can observe that formula_17 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.
The space formula_17 can be endowed with addition, defined as coordinate addition, with a carry bit. That is, for each coordinate, let
formula_20
where formula_21 and
inductively. Increment-by-one is then called the (dyadic) odometer. It is the transformation formula_23 given by formula_24, where formula_2. It is called the "odometer" due to how it looks when it "rolls over": formula_26 is the transformation formula_27. Note that formula_28 and that formula_26 is formula_10-measurable, that is, formula_31 for all formula_32
The transformation formula_26 is non-singular for every formula_34. Recall that a measurable transformation formula_35 is non-singular when, given formula_36, one has that formula_37 if and only if formula_38. In this case, one finds
where formula_40. Hence formula_26 is nonsingular with respect to formula_34. 
The transformation formula_26 is ergodic. This follows because, for every formula_44 and natural number formula_8, the orbit of formula_46 under formula_47 is the set formula_48. This in turn implies that formula_26 is conservative, since every invertible ergodic nonsingular transformation in a nonatomic space is conservative.
Note that for the special case of formula_16, that formula_51 is a measure-preserving dynamical system.
Integer odometers.
The same construction enables to define such a system for every product of discrete spaces. In general, one writes
for formula_53 with formula_54 an integer. The product topology extends naturally to the product Borel sigma-algebra formula_10 on formula_17. A product measure on formula_10 is conventionally defined as formula_58 given some measure formula_59 on formula_60. The corresponding map is defined by 
where formula_62 is the smallest index for which formula_63. This is again a topological group.
A special case of this is the "Ornstein odometer", which is defined on the space 
with the measure a product of
Sandpile model.
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 formula_66 of vertexes and edges. At each vertex formula_67 one places a finite group formula_68 with formula_69 the degree of the vertex formula_70. Transition functions are defined by the graph Laplacian. That is, one can increment any given vertex by one; when incrementing the largest group element (so that it increments back down to zero), each of the neighboring vertexes are incremented by one.
Sandpile models differ from the above definition of a conservative odometer in three different ways. First, in general, there is no unique vertex singled out as the starting vertex, whereas in the above, the first vertex is the starting vertex; it is the one that is incremented by the transition function. 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. 
Markov odometer.
Let formula_71 be an ordered Bratteli–Vershik diagram, consists on a set of vertices of the form formula_72 (disjoint union) where formula_73 is a singleton and on a set of edges formula_74 (disjoint union).
The diagram includes source surjection-mappings formula_75 and range surjection-mappings formula_76. We assume that formula_77 are comparable if and only if formula_78.
For such diagram we look at the product space formula_79 equipped with the product topology. Define "Bratteli–Vershik compactum" to be the subspace of infinite paths,
Assume there exists only one infinite path formula_81 for which each formula_82 is maximal and similarly one infinite path formula_83. Define the "Bratteli-Vershik map" formula_84 by formula_85 and, for any formula_86 define formula_87, where formula_62 is the first index for which formula_89 is not maximal and accordingly let formula_90 be the unique path for which formula_91 are all maximal and formula_92 is the successor of formula_89. Then formula_94 is homeomorphism of formula_95.
Let formula_96 be a sequence of stochastic matrices formula_97 such that formula_98 if and only if formula_99. Define "Markov measure" on the cylinders of formula_95 by formula_101. Then the system formula_102 is called a "Markov odometer".
One can show that the nonsingular odometer is a Markov odometer where all the formula_103 are singletons.