Hi, just a comment, each of the knobs on the navigators panel that you are trying to copy are mechanical memories, non volatile, 1950's style.
The modern version is if you want to stay 360deg rotary will be a rotary quadrature encoder and arduino eprom.
However it means two inputs per rotary encoder, so that clockwise and counter clockwise direction can be determined.
As to scanning the twenty encoders, and to keep the count of arduino inputs low, I don't know.
I have not seen encoders multiplexed and because you need to look at two encoder outputs together and scan them all I'd say its not possible.
However if you give up the idea of a 360 rotary switch and use a 3 position momentary rotary switch that only turns one position left to count down and right from central position to count up, then this can be scanned and arduino eprom used to store the equivalent of the original mechanical memory.
The apperance would be of the original switches, but they only turn probably 1/4 turn clockwise and counter clockwise from a central position, ie 3 position rotary momentary switch.
My ten cents worth.
Tom......... 