标题： 通过随机函数迭代获得的奇异函数 显示英文标题

标题： Singular functions obtained via random function iteration

摘要： 在本文中，我们考虑在实线上通过两个函数的随机迭代产生的离散时间动力系统。 这些函数被假定满足适当的单调性条件；可选地，可以施加对称性条件。 使用二进制序列空间上的伯努利测度，我们证明由迭代过程生成的序列几乎必然发散到正无穷或负无穷。 将每个初始点映射到迭代发散到正无穷的概率的函数被证明满足一个编码自相似性质的泛函方程。 通过这种方式，我们得到了文献中众所周知的奇异函数：类似于康托尔的函数、勒贝格奇异函数以及闵科夫斯基问号函数。 显示英文摘要

摘要： In this paper we consider a discrete-time dynamical system on the real line by random iteration of two functions. These functions are assumed to satisfy appropriate monotonicity conditions; optionally, a symmetry condition may be imposed. Using Bernoulli measures on the space of binary sequences we show that sequences generated by the iteration process almost surely diverge to either plus or minus infinity. The function that assigns to each initial point the probability that the iterates diverge to plus infinity is shown to satisfy a functional equation that encodes self-similarity properties. In this way we obtain singular functions that are well-known from the literature: Cantor-like functions, Lebesgue singular functions, and the Minkowski question mark function.