Entropy of systems

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$ . We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$ .