AN UNCOUNTABLE FAMILY OF GROUP AUTOMORPHISMS, AND A TYPICAL MEMBER

We describe an uncountable family of compact group automorphisms with entropy log 2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical z eta function. If infinitely many Mersenne numbers have a bounded numbe r of prime divisors, then a typical member of the family has upper gro wth rate of periodic points equal to log 2, and lower growth rate equa l to zero.