Parry–Daniels map

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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

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

 

Let π be a permutation such that

 

Then the Parry–Daniels map

 

is defined by

 

References

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  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.