Note on rotation set

Abstract

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