*r*-entropy, equipartition, and Ornstein’s isomorphism theorem in**R**^{n}

## Authors

- Received:

DOI: 10.1007/BF02762054

- Cite this article as:
- Feldman, J. Israel J. Math. (1980) 36: 321. doi:10.1007/BF02762054

- 68 Views

## Abstract

A new approach is given to the entropy of a probability-preserving group action (in the context of**Z** and of**R**_{n}), by defining an approximate “*r*-entropy”, 0<*r*<1, and letting*r* → 0. If the usual entropy may be described as the growth rate of the number of essential names, then the*r*-entropy is the growth rate of the number of essential “groups of names” of width≦*r*, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem for**Z**_{m} ×**R**_{n} and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions of**Z**_{m} ×**R**_{n}, as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.