On the entropy of uniquely ergodic transformations

FRANK HAHNO) AND YITZHAK KATZNELSON(2) 0. Introduction.Ergodic theory involves itself with the study of transformations of a measure space.Topological dynamics is involved with homeomorphisms of a topological space.The entropy of a measure preserving transformation is a gauge of its randomness.There are certain qualitative properties in topological dynamics which seem to play a similar role.For example, equicontinuity, distality, and minimality restrict the "randomness" of a transformation group.The question arises : If a measure preserving transformation is also a homeomorphism is there a strict relation between these notions ?For instance it is known that equicontinuity and distality imply zero entropy.It was not known whether minimality implied zero entropy.W. Parry raised the following question : Let X be a compact metric space and Ta continuous map of Xinto itself.Suppose there is a unique Tinvariant probability measure and suppose (X, T) is minimal (every orbit is dense).Can (X, T) have positive entropy ?The difficulty in answering such a question is the scarcity of a wide class of minimal, uniquely ergodic transformation groups (X, T).We give here a method of constructing minimal uniquely ergodic transformation groups (X, T).Furthermore our method is designed to show that there exist transformation groups (X, T), X a compact metric space, which are minimal and uniquely ergodic but have arbitrarily large finite entropy.Thus minimality and unique ergodicity do not bound the entropy of (X, T) and the answer to Parry's question is no.We carry out our program by constructing closed shift invariant subspaces of a sequence space.In 1 we establish some notation and terminology.In 2 and 3 we either modify or translate some known theorems into combinatorial terms which are to be used in the succeeding sections.In 4 we define the sequences which will be used to build the uniquely ergodic minimal pairs (X, T) of large but finite entropy. 5, 6, 7, 8 are used to carry out the detailed examination of the properties of our sequence.9 indicates the modification necessary in 4 to show that the same construction can be done on bisequerices and thus Tmay also be a homeomorphism.The results are summarized in Corollaries 8.14 and 9.7.The principal idea which makes our result possible is the weak law of large numbers.Lemmas 1.1 and 1.2 are merely forms of this law.The remainder is