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

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