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Journal Article

# Note on Rotation Set

Ryuichi Ito
Proceedings of the American Mathematical Society
Vol. 89, No. 4 (Dec., 1983), pp. 730-732
DOI: 10.2307/2044615
Stable URL: http://www.jstor.org/stable/2044615
Page Count: 3

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Topics: Mathematical theorems, Integers, Mathematical induction, Mathematics

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