Gurevic pressure and equidistribution for amenable extensions of countable state Markov shifts

We obtain a weighted equidistribution theorem for amenable skew product extensions of countable state Markov shifts satisfying the big images and pre-images (BIP) property. We also show, without requiring the BIP property, that Gurevič pressure for an amenable skew product agrees with the Gurevič pressure for the abelianized system. This had been proved by Dougall and Sharp in the case where the base is a subshift of finite type. The equality of Gurevič pressures is a key part of the proof of the equidistribution result.