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A Practical Two-Dimensional Ergodic Theorem
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- Journal:
- Canadian Mathematical Bulletin / Volume 29 / Issue 3 / 01 September 1986
- Published online by Cambridge University Press:
- 20 November 2018, pp. 352-357
- Print publication:
- 01 September 1986
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- Article
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Let τ:[0, 1] → [0, 1] be defined by τ(x) = 2x on [0, 1/2,] and τ(JC) = 2(1 - x) on [5, 1], and let T:[0, 1] x [0, 1] → [0, 1] x [0, 1] be defined by T(x,y) = (τ(x), τ(y))- Let
where p is a prime > 2, and a and M are integers. Consider T restricted to θM x θN, 1 < M < N. Let X = ((2a)/(pM), (2b)/(pN)) ∈ θM x θN and let per(X) denote the length of the period of X.
Then,
where m is Lebesque measure on [0, 1], and C is independent of p, N, M, a and b. Thus, as p → or as N - M and M→ →,
