What is the advantage of the two-part rudder and how does it work? I saw that some aircraft's rudder is composed of an upper rudder and a lower rudder (B727, A380, ...) as illustrated:



Image Source

I wonder how does it work compare to a normal rudder and what are the pros and cons of the two-parts rudder over the one-part rudder?
This is called Split Rudder and it provides redundancy. They run on different systems so if one fails, the other one can be used.

Here is a picture of a split rudder:



Split rudders also provide a finer high speed control, in that only the lower one moves at high speed, reducing the exposed surface area and therefore the control effect. 

Very importantly, there is a structural benefit. When we use the rudder it imposes a twisting load on the vertical fin. By only using the lower rudder, when the aircraft is at high speed, we reduce the twisting moment and transfer that load to a bigger, stronger, part of the airframe. The principle is the same as why the outboard ailerons are disabled at high speed.

The patent information is here.