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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