For a continuous circle map \(f:\mathbb{S}^1 \rightarrow \mathbb{S}^1\) of degree \(d\) with lifting \(F:\mathbb{R} \rightarrow \mathbb{R}\), the author defines the generalized rotation set \(\rho(F,x)\) of a point \(x \in \mathbb{R}\) and the generalized rotation set \(\rho_\Lambda(F)\) on an \(f\)-invariant subset \(\Lambda\) of \(\mathbb{S}^1\) (the generalized rotation set of \(F\) is defined as \(\rho(F)=\rho_{\mathbb{S}^1}(F)\)). When \(d=1\) these definitions give the rotation sets introduced by S. Newhouse et al. [Publ. Math., Inst. Hautes Étud. Sci. 57, 5–71 (1983; Zbl 0518.58031)].
The author’s first main result shows that for a continuous circle map \(f\) with lifting \(F\) such that \(\rho(F,x)\) is a closed subinterval of \(\rho(F)\) for all \(x \in \mathbb{R}\), \(\rho(F)\) is compact and connected. Moreover, there exist two recurrent points of \(f\) such that their generalized rotation numbers are the endpoints of \(\rho(F)\). These results were proven in the case \(d=1\) by R. Bamon et al. [Ergodic Theory Dyn. Syst. 4, 493–498 (1984; Zbl 0605.58027)].
The notion of rotational entropy \(h_r(f)\) was introduced by F. Botelho [Pac. J. Math. 151, No. 1, 1–20 (1991; Zbl 0703.58029)] for an annular map \(f\). The author of the paper reviewed here similarly defines \(h_r(f)\) for a continuous circle map \(f\) and proves that \(h_r(f)=0\).