 In the literature Bernoulli systems are often introduced using so-called shift maps (or Bernoulli shifts). We here briefly indicate how shift maps are related to Bernoulli systems with the example of the baker’s transformation; for a more general discussion see the  Appendix,  Section D. Choose a point in the unit square and write its \(x\) and \(y\) coordinates as binary numbers: \(x = 0.a_1 a_2 a_3\ldots\) and \(y = 0.b_1 b_2 b_3\ldots\), where all the \(a_i\) and \(b_i\) are either 0 or 1. Now put both strings together back to back with a dot in the middle to form one infinite string: \(S= \ldots b_3 b_2 b_1 .a_1 a_2 a_3\ldots\), which may represent the state of the system just as a ‘standard’ two-dimensional vector does. Some straightforward algebra then shows that