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Iteration of quadrilateral foldings. (Itération de pliages de quadrilatères.) (French) Zbl 1073.37002

Let \(q_{0}\) be a quadrilateral of \(\mathbb{R}^{2}\) such that its sidelengths are \(a_{1},a_{2},a_{3},a_{4}\). We assume that \((a_{1},a_{2})\) is different from \((a_{4},a_{3})\) and \((a_{1},a_{4})\) is different from \((a_{2},a_{3})\) which implies that the quadrilaterals \(q_{n}=\phi_{4}\circ \phi_{3}\circ \phi_{2}\circ \phi_{1}(q_{n-1})\) obtained by iteration of foldings (where the folding \(\phi_{j}\) replaces the vertex number \(j\) by its symmetric with respect to the opposite diagonal) is well defined. By means of numerical experimentations, Charter and Rogers described some phenomena about the dynamical behavior of the sequence \((q_{n})\) when \(a_{1}=a_{3}\), i.e., for isosceles quadrilaterals and they conjectured that the sequence \((q_{n})\) is bounded whenever \(a_{2}+a_{4}\leq p\) where \(p\) denotes the semiperimeter of the quadrilateral (for details see K. Charter and T. Rogers [Exp. Math. 2, 209–222 (1993; Zbl 0802.58035)]).
The goal of the paper under review is to show the validity of these phenomena and to give an answer to Charter-Rogers’ conjecture. As an example of the results stated in the paper we have: Theorem 1. The drift \(v(q_{0})=\lim_{n\rightarrow \infty}(1/n)q_{n}\) exists. Theorem 2. If \(q_{0}\) is of nonperiodic type, then \(v(q_{0})=0\) if and only if \(\max_{1\leq j\leq 4}\{a_{j}\}+\min_{1\leq j\leq 4}\{a_{j}\} \leq (1/2)\sum_{1\leq j\leq 4}a_{j}\). Theorem 3. If \(q_{0}\) is of nonperiodic type, then the sequence \((q_{n})\) converges to infinite if and only if \(v(q_{0})\) is different from zero. In relation to Charter-Rogers’ conjecture it is shown that it is true in the case that the vertices of \(q_{0}\) are algebraic, and that it is also true for Lebesgue almost all \(q_{0}\) but that is false for Baire generic \(q_{0}\).

MSC:

37A05 Dynamical aspects of measure-preserving transformations
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28D05 Measure-preserving transformations
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37B99 Topological dynamics

Citations:

Zbl 0802.58035
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Full Text: DOI

References:

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