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. It is named after the England mathematician Bill Parry and the British statistician Henry Daniels, 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

[math]\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 \}. }[/math]

Let π be a permutation such that

[math]\displaystyle{ x_{\pi(0)} \leq x_{\pi (1)} \leq \dots \leq x_{\pi (n)}. }[/math]

Then the Parry–Daniels map

[math]\displaystyle{ T_{\pi} : \Sigma \to \Sigma }[/math]

is defined by

[math]\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). }[/math]