Abstract
Throughout this monograph the following conventions are adopted:
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The circle is represented as the quotient \({\mathbb T} = {\mathbb R}/{\mathbb Z}\).
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\(\pi : {\mathbb R} \to {\mathbb T}\) is the canonical projection.
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Three or more distinct points \(t_1, t_2, \ldots , t_k \in {\mathbb T}\) are in positive cyclic order if there are representatives x i  ∈ π −1(t i ) such that x 1 < x 2 < ⋯ < x k  < x 1 + 1.
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For a distinct pair \(t_1,t_2 \in {\mathbb T}\), the interval \((t_1,t_2) \subset {\mathbb T}\) is defined as the set of all \(t \in {\mathbb T}\) such that t 1, t, t 2 are in positive cyclic order. We define the intervals (t 1, t 2], [t 1, t 2), [t 1, t 2] by adding the suitable endpoints to (t 1, t 2).
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The length of an interval \((t_1,t_2) \subset {\mathbb T}\) is always understood as its normalized Lebesgue measure, that is, the unique representative of t 2 − t 1 in [0, 1).
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Notes
- 1.
Let us emphasize that our plateaus are open intervals, a convention that is not commonly adopted in the literature.
References
A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1995)
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Zakeri, S. (2018). Monotone Maps of the Circle. In: Rotation Sets and Complex Dynamics. Lecture Notes in Mathematics, vol 2214. Springer, Cham. https://doi.org/10.1007/978-3-319-78810-4_1
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DOI: https://doi.org/10.1007/978-3-319-78810-4_1
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