Kneading theory and rotation intervals for a class of circle maps of degree one. (English) Zbl 0735.54026

Let \(e: \mathbb R\to S^1\) be the projection associating with \(x\) the complex number \(\exp(2\pi ix)\). A continuous mapping \(F: \mathbb{R} \to \mathbb{R}\) is a lifting of a continuous map \(f: S^1 \to S^1\) if \(e\circ F=f\circ e\). An integer \(k\) such that \(f(x + 1)=F(x) + k\) for all \(x\) is called the degree of \(F\). The class of all liftings of continuous maps of the circle into itself of degree one is denoted by \(\mathcal L\). If \(F\in\mathcal L\) and \(x\in \mathbb{R}\) then \(\varrho_ F(x):= \limsup (F^n(x)-x)/n\) as \(n\to \infty\) is called the rotation number of \(F\) for \(x\), following S. Newhouse, J. Palis and F. Takens [Publ. Math., Inst. Hautes Étud. Sci. 57, 5–71 (1983; Zbl 0518.58031)]. The set of all rotation numbers of \(F\) is denoted by \(L_F\), being a closed interval on \(\mathbb{R}\) as is proved by R. Ito in [Math. Proc. Camb. Philos. Soc. 89, 107–111 (1981; Zbl 0484.58027)], and is called the rotation interval of \(F\). For \(F\in\mathcal L\), \(F\) being a lifting of \(f\), the topological entropy of \(F\) is defined, following R. L. Adler, A. G. Konheim and M. H. MacAndrew [Trans. Am. Math. Soc. 114, 309–319 (1965; Zbl 0127.13102)]; M. Denker, C. Grillenberger and it{K. Sigmund} [Ergodic theory on compact spaces, Lect. Notes Math. 527 (1976; Zbl 0328.28008)], as the topological entropy of \(f\).
The author investigates a class \(\mathcal A\) of maps defined as follows: \(F\in \mathcal A\) iff (1) \(F \in \mathcal L\), (2) there exists \(c_F\in (0,1)\) such that \(F\) is strictly increasing in \((0,c_F)\) and strictly decreasing in \((c_F,1)\). For the class \(\mathcal A\) the author proposes a kneading theory. For a map belonging to \(\mathcal A\) he characterizes the set of itineraries depending on the rotation interval. From this result he gets lower and upper bounds of the topological entropy and of the number of periodic orbits of each period. These lower bounds appear to be valid for a general continuous map of the circle of degree one.


37B40 Topological entropy
37E10 Dynamical systems involving maps of the circle
37E45 Rotation numbers and vectors
54H20 Topological dynamics (MSC2010)
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