mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Parry–Daniels map is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

studied in the context of

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...

. Typical questions concern the existence of an

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

ergodic measure 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 t ...

for the map. It is named after the

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Bill Parry William or Bill Parry may refer to: Sports *William Parry Crake (1852–1921), or William Parry, Wanderers footballer *Bill Parry (footballer, born 1873) (1873–1923), Welsh international footballer *Bill Parry (footballer, born 1914) (1914–196 ...

and the

British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity and common culture * British English, ...

statistician A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...

Henry Daniels Henry Ellis Daniels FRS (2 October 1912 – 16 April 2000) was a British statistician. He was President of the Royal Statistical Society (1974–1975), and was awarded its Guy Medal in Gold in 1984, following a silver medal in 1947. He bec ...

, who independently studied the map in papers published in 1962.

Definition

Given an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

''n'' ≥ 1, let Σ denote the ''n''-

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

in R^''n''+1 given by :

\Sigma := \.

Let ''π'' be a

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

such that :

x_ \leq x_ \leq \dots \leq x_.

Then the Parry–Daniels map :

T_ : \Sigma \to \Sigma

is defined by :

T_\pi (x_0, x_1, \dots, x_n) := \left( \frac , \frac, \dots, \frac \right).

References

Dynamical systems {{mathanalysis-stub