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A Practical Two-Dimensional Ergodic Theorem

Published online by Cambridge University Press:  20 November 2018

Manny Scarowsky  and
Abraham Boyarsky
Manny Scarowsky
Affiliation:
Department of MathematicsLoyola Campus, Concordia University7141 Sherbrooke ST. West Montreal, CanadaH4B 1R6
Abraham Boyarsky
Affiliation:
Department of MathematicsLoyola Campus, Concordia University7141 Sherbrooke ST. West Montreal, CanadaH4B 1R6
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Abstract

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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→ →,

Keywords

28D1026A1
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 3 , 01 September 1986 , pp. 352 - 357
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Lasota, A. and Yorke, J. A., 0A? the existence of invariant measures for piecewise monotonie transformations, Trans. Amer. Math. Soc, 183 (1973), pp. 481–485.Google Scholar
2. Scarowsky, M. and Boyarsky, A., Long periodic orbits of the triangle map, Proc. Amer. Math. Soc. (in press).Google Scholar