Parry–Daniels map

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

${\displaystyle \Sigma :=\{x=(x_{0},x_{1},\dots ,x_{n})\in \mathbb {R} ^{n+1}|0\leq x_{i}\leq 1{\mbox{ for each }}i{\mbox{ and }}x_{0}+x_{1}+\dots +x_{n}=1\}.}$

Let π be a permutation such that

${\displaystyle x_{\pi (0)}\leq x_{\pi (1)}\leq \dots \leq x_{\pi (n)}.}$

Then the Parry–Daniels map

${\displaystyle T_{\pi }:\Sigma \to \Sigma }$

is defined by

${\displaystyle T_{\pi }(x_{0},x_{1},\dots ,x_{n}):=\left({\frac {x_{\pi (0)}}{x_{\pi (n)}}},{\frac {x_{\pi (1)}-x_{\pi (0)}}{x_{\pi (n)}}},\dots ,{\frac {x_{\pi (n)}-x_{\pi (n-1)}}{x_{\pi (n)}}}\right).}$

References

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