For transmit, the voltage at the center of a dipole not cut in the center and fed with coax and a gamma match is zero in the center at the resonant frequency. This is why apparently a 'plumbers delight' yagi with the driven element connected to the boom works ok, because the voltage in the middle is zero.
And then if the voltage is zero in the middle you don't have to worry about common mode current flowing down the outside of the coax because at the point where the outer of the coax joins to the middle of the driven element the voltage is zero and no current can flow anywhere.
So what about when you move the frequency of the signal applied to the feed point away from resonance, then :
Do you get current on the outside of the coax now because now the voltage in the middle isn't zero anymore ?
And if you are outputting harmonics then will these result in corresponding current on the outside of the coax because voltage in the antenna caused by the harmonics are not at the resonant frequency and so that voltage won't be zero in the middle at the harmonic frequencies ?
And also does this all help to make the antenna want to have a narrow bandwidth ?