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 , 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. [2] These are sometimes referred to as adding machines. [3] 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. [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. [3]
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 Bernoulli measures, given by and , for some independent of . The value of is rather special; it corresponds to the special case of the Haar measure, when 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 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 can be endowed with addition, defined as coordinate addition, with a carry bit. That is, for each coordinate, let where and
inductively. Increment-by-one is then called the (dyadic) odometer. It is the transformation given by , where . It is called the odometer due to how it looks when it "rolls over": is the transformation . Note that and that is -measurable, that is, for all
The transformation is non-singular for every . Recall that a measurable transformation is non-singular when, given , one has that if and only if . In this case, one finds
where . Hence is nonsingular with respect to .
The transformation is ergodic. This follows because, for every and natural number , the orbit of under is the set . This in turn implies that is conservative, since every invertible ergodic nonsingular transformation in a nonatomic space is conservative.
Note that for the special case of , that is a measure-preserving dynamical system.
The same construction enables to define such a system for every product of discrete spaces. In general, one writes
for with an integer. The product topology extends naturally to the product Borel sigma-algebra on . A product measure on is conventionally defined as given some measure on . The corresponding map is defined by
where is the smallest index for which . 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
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 of vertexes and edges. At each vertex one places a finite group with the degree of the vertex . 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.
Let be an ordered Bratteli–Vershik diagram, consists on a set of vertices of the form (disjoint union) where is a singleton and on a set of edges (disjoint union).
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 singletons.
In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of an unspecified person from within a group would be a random vector. Normally each element of a random vector is a real number.
In probability and statistics, a Bernoulli process 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. The component Bernoulli variablesXi are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin. Every variable Xi in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ; this generalization is known as the Bernoulli scheme.
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics as well as systems in thermodynamic equilibrium.
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: e.g. mixing paint, mixing drinks, industrial mixing.
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function.
The dyadic transformation is the mapping
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal.
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system X being in state x as
In mathematics, a π-system on a set is a collection of certain subsets of such that
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”
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 that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.
In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure onto a finite-dimensional vector space.
In mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and
In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson & Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. They give an 'coordinate independent' notion of vertex algebras, which are based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras.
In machine learning, the kernel embedding of distributions comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space on which a sensible kernel function may be defined. For example, various kernels have been proposed for learning from data which are: vectors in , discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.
A Markov chain on a measurable state space is a discrete-time-homogeneous Markov chain with a measurable space as state space.
A generative adversarial network (GAN) is a class of machine learning frameworks and a prominent framework for approaching generative AI. The concept was initially developed by Ian Goodfellow and his colleagues in June 2014. In a GAN, two neural networks contest with each other in the form of a zero-sum game, where one agent's gain is another agent's loss.
In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory.