Not sure about the Jack antenna but for the batwing style, the only wire from the little box power supply and the antenna itself is the coaxial cable. The power to the antenna uses the centre connector and the shield of the coax to power the amp in the head of the antenna itself. The TV signal from the antenna used the same coax cable. There are no splitters in that portion of the cable in most cases.
In many cases the coax cable from the park cable entrance also goes to that same power supply. When the power is on (button pushed in) the coax is effectively connected from the antenna to the TV. When the button is off, the Cable connection is connected through to the TV. The portion of the coax from the TV to the little box is common to both antenna and cable connections. The little box also has a fused 12V and ground wires connection to it.
You should be able to dismount the little box and check the system there.
I hope all this also applies to the Jack antenna as well!!