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 English 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
Σ := { x = ( x 0 , x 1 , … , x n ) ∈ R n + 1 | 0 ≤ x i ≤ 1 for each i and x 0 + x 1 + ⋯ + x n = 1 } . {\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
x π ( 0 ) ≤ x π ( 1 ) ≤ ⋯ ≤ x π ( n ) . {\displaystyle x_{\pi (0)}\leq x_{\pi (1)}\leq \dots \leq x_{\pi (n)}.}
Then the Parry–Daniels map
T π : Σ → Σ {\displaystyle T_{\pi }:\Sigma \to \Sigma }
is defined by
T π ( x 0 , x 1 , … , x n ) := ( x π ( 0 ) x π ( n ) , x π ( 1 ) − x π ( 0 ) x π ( n ) , … , x π ( n ) − x π ( n − 1 ) x π ( n ) ) . {\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).}
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