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Skew-products with simple approximations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Whitman, P. N. |
| Copyright Year | 1978 |
| Abstract | Conditions are given in order that the cartesian product of two measure-preserving invertible transformations admits an approximation. A class of skew-product transformations is defined and conditions are given for a member of this class to admit a simple approximation. 1. Preliminaries. Let (X, 'C, ,u) be a Lebesgue space; that is, a measure space isomorphic to the unit interval with Lebesgue measure. A measurepreserving invertible point transformation of X is called an automorphism of (X, ', 1)4 Let T: X -> X be an automorphism. The induced automorphism TA: A -* A where A c f is defined as follows: TAx-Tkx, xEEA where k is the least positive integer such that Tkx E A. Let Z denote the set of positive integers. Let f: X -Z be an integrable function. The special automorphism over T built under the function f is defined as follows: Put B(k, n) = {(x, n): x E X, f (x) = k}, n, k E Z and 1 l U l l B(k, 1). We may regard each set B(k, n), 1 < n < k, as a copy of B(k, 1). Consequently we may extend the measure ,t to X(J) and form a normalised measure /L' on X(J) in the obvious way. We define Tf, the special automorphism over T, by Tf(x,n)=(x,n+ 1) if I n |
| Starting Page | 521 |
| Ending Page | 526 |
| Page Count | 6 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1978-0509247-5 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1978-072-03/S0002-9939-1978-0509247-5/S0002-9939-1978-0509247-5.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1978-0509247-5 |
| Volume Number | 72 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
