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Circle homeomorphisms with weak discontinuities. (English) Zbl 0733.58026
Dynamical systems and statistical mechanics, Pap. Semin. Stat. Phys., Moscow/USSR, Adv. Sov. Math. 3, 57-98 (1991).

[For the entire collection see Zbl 0723.00030.]

The authors consider one-parameter families of mappings of the unit circle: $T{f}_{{\Omega }}\left(x\right)=the$ fractional part of ${f}_{{\Omega }}\left(x\right)$, $x\in {S}^{1}$, where ${f}_{{\Omega }}\left(y\right)=f\left(y\right)+{\Omega }$, $y\in {ℝ}^{1}$, $0\le {\Omega }<1$. Here $f\left(0\right)=0$, $f\left(y+1\right)=f\left(y\right)+1,$ $y\in {ℝ}^{1}$ and f(y) is assumed to satisfy some other conditions which mean that ${f}_{{\Omega }}\left(x\right)$ defines a homeomorphism of the circle which is sufficiently smooth except for a single point where a weak discontinuity (a jump of the first derivative) occurs.

The use of renormalization group techniques allows the authors to prove some important properties of such homeomorphisms. In particular, they show that for any rational rotation number p/q there exist at most two periodic trajectories of period q. Further, they prove that the parameter values corresponding to irrational rotation numbers form a set of zero Lebesgue measure [for critical mappings of the circle this was proved by G. Świa̧tek, Commun. Math. Phys. 119, No.1, 109-128 (1988; Zbl 0656.58017)].

##### MSC:
 37B99 Topological dynamics