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 zeta function.

If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log 2, and lower growth rate equal to zero.