\(\rho \)-bounded orbits and minimal sets for generalized quasiperiodically forced circle maps

We study the generalized quasiperiodically forced circle map \(f:\mathbb {T}^{m}\times \mathbb {T}^{1}\rightarrow \mathbb {T}^{m}\times \mathbb {T}^{1}\), which is a natural generalization of the quasiperiodically forced circle map. Our main aim is to show that for each \(\rho \) in the interior of the fibred rotation set, there is a minimal set such that each orbit on the minimal set is \(\rho \)-bounded.