Abstract
In this paper, we consider the generalized rotation set and rotational entropy for circle maps. Let f:S1→S1 be continuous and F be a lifting of f. In [1] a notion of F-rotation set of a point x∈S1 was given when f is homotopic to the identity. We firstly extend this notion, designated ‘generalized rotation set’ ρ(F), to the general continuous map of the circle, not necessarily homotopic to identity. We show that ρ(F) is compact and connected, i.e., a closed interval, and moreover, each of its endpoints is a generalized rotation number of a recurrent point of f. We also consider the generalized rotation set for f on its invariant sets. Secondly, we introduce a notion of generalized rotational entropy hr(f) for f, which follows the definition that Botelho proposed in [3] for the annulus, and then we show that it exactly vanishes.
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