Cascades in the dynamics of affine interval exchange transformations. (English) Zbl 1448.37047
An affine interval exchange transformation is a piecewise continuous bijection of the unit interval that is affine when restricted to its intervals on continuity. Such transformations can be quite complicated. The possibilities include asymptotic periodicity, minimality or – unusually – a map with an invariant quasi-minimal Cantor set.
The authors begin by defining an affine interval exchange transformation \(F\) on the half-open interval \(D = [0,1)\) by \(F(x) = 2x + \frac{1}{6}\) for \(x\) in \([0, \frac{1}{6})\), \(F(x) = \frac{1}{2} (x - \frac{1}{6} )\) for \(x\) in \([\frac{1}{6}, \frac{1}{2})\), \(F(x) = \frac{1}{2} (x - \frac{1}{2} ) + \frac{5}{6}\) for \(x\) in \([\frac{1}{2}, \frac{5}{6})\), and \(F(x) = 2 (x - \frac{5}{6} ) + \frac{1}{2}\) for \(x\) in \([\frac{5}{6}, 1)\). Note that \(F^2 (x) = x\).
The one-parameter family \(F_t\), parametrized by \(t\) in \(S^1\), is then defined by \(F_t (x)= F(r_t(x))\) where \(r_t\) is translation by \(t\) modulo 1 on \([0, 1)\). The authors’ first theorem claims that \(F_t\) is dynamically trivial for almost all \(t\) in \(S^1\). This means that there are two periodic points \(x^+\) and \(x^-\) in \(D\) of integer orders \(p\) and \(q\) such that \((F{_t}^p)^\prime (x^+) < 1\), \((F{_t}^q)^\prime (x^-) > 1\), and for all \(z\) not in the orbit of \(x^-\), the \(\omega\) limit set of \(z\) is the orbit of \(x^+\). One of tasks in this paper is to show that this one-parameter family displays a very rich dynamical behavior.
The authors also prove the following theorem. For all \(t\) in a Cantor set of parameters in \(S^1\) there is a Cantor subset \(C_t\) of \(D\) such that for all \(x\) in \(D\), the \(\omega\)-limit set of \(x\) under \(F_t\) is equal to \(C_t\).
A key aspect of the paper is the correspondence of the family consisting of \(\{F_t : t \in S^1\}\) with a dilatation surface \(\Sigma\). The dilation surface \(\Sigma\) is naturally endowed with a family of foliations that the authors call directional foliations. Hidden symmetries appear because the surface \(\Sigma\) has a group of affine symmetries that is non-trivial. This eventually leads to an analysis of the Veech group associated with the dilatation surface \(\Sigma\).
The authors begin by defining an affine interval exchange transformation \(F\) on the half-open interval \(D = [0,1)\) by \(F(x) = 2x + \frac{1}{6}\) for \(x\) in \([0, \frac{1}{6})\), \(F(x) = \frac{1}{2} (x - \frac{1}{6} )\) for \(x\) in \([\frac{1}{6}, \frac{1}{2})\), \(F(x) = \frac{1}{2} (x - \frac{1}{2} ) + \frac{5}{6}\) for \(x\) in \([\frac{1}{2}, \frac{5}{6})\), and \(F(x) = 2 (x - \frac{5}{6} ) + \frac{1}{2}\) for \(x\) in \([\frac{5}{6}, 1)\). Note that \(F^2 (x) = x\).
The one-parameter family \(F_t\), parametrized by \(t\) in \(S^1\), is then defined by \(F_t (x)= F(r_t(x))\) where \(r_t\) is translation by \(t\) modulo 1 on \([0, 1)\). The authors’ first theorem claims that \(F_t\) is dynamically trivial for almost all \(t\) in \(S^1\). This means that there are two periodic points \(x^+\) and \(x^-\) in \(D\) of integer orders \(p\) and \(q\) such that \((F{_t}^p)^\prime (x^+) < 1\), \((F{_t}^q)^\prime (x^-) > 1\), and for all \(z\) not in the orbit of \(x^-\), the \(\omega\) limit set of \(z\) is the orbit of \(x^+\). One of tasks in this paper is to show that this one-parameter family displays a very rich dynamical behavior.
The authors also prove the following theorem. For all \(t\) in a Cantor set of parameters in \(S^1\) there is a Cantor subset \(C_t\) of \(D\) such that for all \(x\) in \(D\), the \(\omega\)-limit set of \(x\) under \(F_t\) is equal to \(C_t\).
A key aspect of the paper is the correspondence of the family consisting of \(\{F_t : t \in S^1\}\) with a dilatation surface \(\Sigma\). The dilation surface \(\Sigma\) is naturally endowed with a family of foliations that the authors call directional foliations. Hidden symmetries appear because the surface \(\Sigma\) has a group of affine symmetries that is non-trivial. This eventually leads to an analysis of the Veech group associated with the dilatation surface \(\Sigma\).
Reviewer: William J. Satzer Jr. (St. Paul)
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 37E35 | Flows on surfaces |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
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