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Persistent rotation intervals for old maps. (English) Zbl 0714.58030
Dynamical systems and ergodic theory, 28th Sem. St. Banach Int. Math. Cent., Warsaw/Pol. 1986, Banach Cent. Publ. 23, 119-124 (1989).
[For the entire collection see Zbl 0686.00015.]
Let us call o.l.d map the continuous lifting to $${\mathbb{R}}$$ of a continuous map of degree one of the circle onto itself. Such a lifting satisfies $$f(x+1)=f(x)+1$$ for any $$x\in {\mathbb{R}}$$. For an o.l.d map one defines its rotation interval L(f) by $L(f)=closure\{\limsup_{n\to \infty}(1/n)(f^ n(x)-x);\quad x\in {\mathbb{R}}\}.$ Let $${\mathcal A}$$ be a topological vector space of o.l.d maps. A function $$f\in {\mathcal A}$$ is said to have the property of persistent rotation interval if there exists a neighbourhood U of f in $${\mathcal A}$$ such that $$L(g)=L(f)$$ for any $$g\in U.$$
The essential problem discussed in this paper is to give sufficient conditions on $${\mathcal A}$$ implying that the subset of functions with persistent rotation interval is dense in $${\mathcal A}$$. This family of t.v.s. contains the spaces $$C^ r$$, $$r=0,1,...,\infty$$ of smooth o.l.d functions as well as the set of real analytic o.l.d functions.
Reviewer: J.Lacroix

##### MSC:
 37A99 Ergodic theory
##### Keywords:
old map; rotation interval