1986 Mazda RX7 1986 RX7 sometimes will not start

Our 1986 RX-7 has problems starting sometimes. The engine has no known mechanical problems and once started, it performs just great and we can stop and turn the engine off and it will start agin with ease the next day. However, there are times when we can not start the engine the next day. The battery has been replaced. When we attempt to start the car, nothing happens and the engine does not even turn. We also replaced the ignition assembly and cables that are directly behind the key on the steering column. But this did not seem to help. Are there any known problems with Mazda 1986 RX7 of these nature?
Thanks for your help.
Don in Iowa