Rotation intervals for a class of maps of the real line into itself. (English) Zbl 0615.54030

Misiurewicz, Michał

Ergodic Theory Dyn. Syst. 6, 117-132 (1986).

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 \(p/q\).

Reviewer: Gh.Toader

MSC:

54H20

Topological dynamics (MSC2010)

Keywords:

degree one liftings of maps of the circle; discontinuities; mod 1 maps; maps arising from Newton’s method; heavy maps; periodic point; rotation number

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