Note on rotation set

Abstract:

Let $f$ be an endomorphism of the circle of degree 1 and $\bar f$ be a lifting of $f$. We characterize the rotation set $\rho (\bar f)$ by the set of probability measures on the circle, and prove that if ${\rho _ + }(\bar f)\;({\rho _ - }(\bar f))$, the upper (lower) endpoint of $\rho (\bar f)$, is irrational, then ${\rho _ + }({R_\theta }\bar f) > {\rho _ + }(\bar f)\;({\rho _ - }({R_\theta }\bar f) > {\rho _ - }(\bar f))$ for any $\theta > 0$, where ${R_\theta }(x) = x + \theta$. As a corollary, if $f$ is structurally stable, then both ${\rho _ + }(\bar f)$ and ${\rho _ - }(\bar f)$ are rational.