Parry–Daniels map explained

In mathematics, the Parry–Daniels map is a function studied in the context of dynamical systems. Typical questions concern the existence of an invariant or ergodic measure for the map.^[1]

It is named after the English mathematician Bill Parry^[2] and the British statistician Henry Daniels,^[3] who independently studied the map in papers published in 1962.

Definition

Given an integer n ≥ 1, let Σ denote the n-dimensional simplex in Rⁿ⁺¹ given by

\Sigma : = \{ x = (x _{0,} x _{1,} ... , x _{n)} \in R ^{n} | 0 \leq x _{i} \leq 1 foreach i and x _{0} + x _{1} + ... + x _{n} = 1 \}.

Let π be a permutation such that

x _{\pi(0)} \leq x _{\pi} \leq ... \leq x _{\pi} .

Then the Parry–Daniels map

T _{\pi} : \Sigma \to \Sigma

is defined by

T _{\pi} (x _{0,} x _{1,} ... , x _{n)} : = \left( (x _{\pi})/(x _{\pi}) , (x _{\pi} - x _{\pi})/(x _{\pi}) , ... , (x _{\pi} - x _{\pi})/(x _{\pi}) \right).

Notes and References

1. Web site: Zweimüller . Roland . Surrey Notes on Infinite Ergodic Theory .
2. Parry . William . 1962 . Ergodic Properties of Some Permutation Processes . Biometrika . 49 . 1/2 . 151–154 . 10.2307/2333475 . 2333475 . 0006-3444.
3. Daniels . H. E. . 1962 . Processes Generating Permutation Expansions . Biometrika . 49 . 1/2 . 139–149 . 10.2307/2333474 . 2333474 . 0006-3444.