Linear fractional transformations mod one and ergodic theory

• Main Author: Rudolpher, Stephan Martin
• Published: Imperial College London 1968
• Subjects: 510
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.623122

After an introductory chapter, we study characterisations of mixing, weak mixing and ergodicity of a finite measure-preserving transformation T due to N. Oishi [25]. These characterisations are in terms of convergence of suitably defined entropies of finite partitions. We show that the characterisations can be given in terms of (countable) partitions with finite entropy, extend the characterisation to mixing of degree r and give further characterisations in terms of convergence of the suitably defined measures of Jordan measurable sets and, in the case of a compact measure space, in terms of weak convergence of these measures. It is shown that these characterisations cannot be extended to convergence of the corresponding entropies of TxT nor to all measurable subsets, respectively. Chapter III studies the ergodic properties of two classes of linear fractional transformation mod one, which turn out mostly to have similar properties to previously studied f-transformations [29], [32]. The main tool is a sufficient condition for ergodicity of non-singular, many-one transformations of a probability space, which, applied to f-transformations, generalises a similar theorem of A. Renyi [29]. Renyi's theorem states the existence of a finite invariant measure equivalent to Lebesgue measure. In some cases, using a result of W. Parry [27], we have succeeded in constructing this invariant measure. Throughout, results were only obtained for f-transformations with independent digits (in the sense of Renyi). The dependent digit case is much more delicate, and we were unable to obtain results in this direction.