If the gates of the moon are ruled by a cadet branch of the royce , does not it mean that the arryns have no lands of their own ?

No. For one thing, the Gates of the Moon are a single castle. Not only do the Arryns have all the castles of the Eyrie, but they have extensive lands in the Vale of Arryn behind it. 

For another, up until Petyr Baelish became Lord Protector, the Royces were not hereditary rulers of anything, but rather servants of the Arryns in the same way that the Tyrells served the Gardeners or the Cassels serve the Starks.