随机系统的匹配及其在极小权展开中的应用
Matching for random systems with an application to minimal weight expansions
摘要：
我们将一维动力系统的匹配概念推广到区间上随机动力系统的随机匹配。证明了对于区间上的一个大的分段仿射随机系统族，随机匹配的性质意味着任何不变密度都是分段常数。我们进一步引入了一个单参数随机动力系统族，该系统产生区间[-1，1]中的数字的有符号二进制展开式。这个族几乎每个参数都是随机匹配的。我们用它来证明数字0在相关的有符号二进制展开式中的频率永远不会超过1/2。
英文摘要：
We extend the notion of matching for one-dimensional dynamical systems to random matching for random dynamical systems on an interval. We prove that for a large family of piecewise affine random systems of the interval the property of random matching implies that any invariant density is piecewise constant. We further introduce a one-parameter family of random dynamical systems that produce signed binary expansions of numbers in the interval [-1,1]. This family has random matching for Lebesgue almost every parameter. We use this to prove that the frequency of the digit 0 in the associated signed binary expansions never exceeds 1/2.
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