In mathematics, a Markov odometer is a certain type of
topological dynamical system In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
Scope
The central object of study in topolog ...
. It plays a fundamental role in
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
and especially in
orbit theory of dynamical systems, since a theorem of
H. Dye asserts that every
ergodic
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
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
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
defined on the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
of
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
s, induced by addition defined as
, where
. This group can be endowed with the structure of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
; the result is a
conservative dynamical system.
The general form, which is called "Markov odometer", can be constructed through
Bratteli–Vershik diagram In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (''V'', ''E'') with a homeomorphism on the set of all infinite paths called the Veršhik transformation. It is named after Ola Bratteli and Anatoly ...
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
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. Th ...
. This is the set of all infinite strings in two symbols, here denoted by
endowed with the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. This definition extends naturally to a more general odometer defined on the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
:
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
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
has a natural topology, the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
, generated by the
cylinder set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.
General definition
Given a collection S of sets, consider the Cartesian product X = \prod ...
s. 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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Measu ...
, the Bernnoulli measures, given by
and
, for some