高维连分数的偶然分解
Serendipitous decompositions of higher-dimensional continued fractions
摘要：
我们证明了一套动力学的结果，包括精确的变换和分段解析的不变测度，为一个家庭的连分数系统，包括具体的例子，实数，复数，四元数，八元数，并在$R^3$。我们的方法扩展了中田和亨斯利的工作，特别是填补了一些空白，亨斯利的分析赫尔维茨复杂的连分数。我们进一步为连分数算法引入了一个新的“偶然性”条件，该条件控制了基本区边界在连分数映射迭代下的长期行为，并且在合理的条件下等价于有限值域性质.我们还表明，有限范围的条件是非常微妙的：由非二次无理数的偶然系统的扰动不保持偶然，实验证据表明，偶然可能会失败，甚至一些合理的扰动。
英文摘要：
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers, quaternions, octonions, and in $R^3$. Our methods expand on the work of Nakada and Hensley, and in particular fill some gaps in Hensley's analysis of Hurwitz complex continued fractions. We further introduce a new ``serendipity'' condition for a continued fraction algorithm, which controls the long-term behavior of the boundary of the fundamental domain under iteration of the continued fraction map, and which is under reasonable conditions equivalent to the finite range property. We also show that the finite range condition is extremely delicate: perturbations of serendipitous systems by non-quadratic irrationals do not remain serendipitous, and experimental evidence suggests that serendipity may fail even for some rational perturbations.
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