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Rotation intervals for a class of maps of the real line into itself. (English) Zbl 0615.54030
A map $$F: {\mathbb{R}}\to {\mathbb{R}}$$ is called an old map (old stands for ”degree one lifting”) if $$F(X+1)=F(X)+1$$ for all $$X\in {\mathbb{R}}$$. A point X is called periodic mod 1 of period q and rotation number p/q for an old map F if $$F^ q(X)-X=p$$ and $$F^ i(X)-X\not\in Z$$ for $$i=1,2,...,q-1$$. The author defines for an old map $$F:$$ $a(F)=\inf_{X\in {\mathbb{R}}} \liminf_{n\to \infty}(F^ n(X)-X)/n\text{ and } b(F) = \sup_{X\in {\mathbb{R}}} \limsup_{n\to \infty}(F^ n(X)-X)/n.$ A map $$F: {\mathbb{R}}\to {\mathbb{R}}$$ is called heavy if it has one-sided limits, F(X-) and $$F(X+)$$, in every point X and $$F(X-)\geq F(X)\geq F(X+)$$. The author generalizes some results on continuous maps to heavy maps. We give for example: Theorem A. Let F be an old heavy map. Then (a) if F has a periodic point of rotation number $$p/q$$ then $$a(F)\leq p/q\leq b(F)$$; (b) if $$a(F)<p/q<b(F)$$ then F has a periodic mod 1 point of period q and rotation number $$p/q$$.