CHAOS CAUSED BY A STRONG-MIXING MEASURE-PRESERVING TRANSFORMATION

The chaos caused by a strong-mixing preserving transformation is discu ssed and it is shown that for a topological space X satisfying the sec ond axiom of countability and for an outer measure m on X satisfying t he conditions: (i) every non-empty open set of X is m-measurable with positive m-measure; (ii) the restriction of m on Borel sigma-algebra B (X) of X is a probability measure, and (iii) for every Y subset of X t here exists a Borel set B subset of B(X) such that B superset of Y and m(B) = m(Y), if f:X-->X is a strong-mixing measure-preserving transfo rmation of the probability space (X, B(X), m), and if i mi I is a stri ctly increasing sequence of positive integers, then there exists a sub set C subset of X with m(C) = 1, finitely chaotic with respect to the sequence {m(i)}, i. e. for any finite subset A of C and for any map F: A-->X there is a subsequence {r(i)} such that lim(i-->infinity)f(r)i(a ) = F(a) for any a is an element of A. There are some applications to maps of one dimension.