Power Domains and Iterated Function Systems
Power Domains and Iterated Function Systems
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DOI：
10.1006/inco.1996.0014
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发表时间：
1996-02
期刊：
影响因子：
--
通讯作者：
A. Edalat
中科院分区：
文献类型：
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作者：
A. Edalat
引入紧度量空间上弱双曲迭代函数系统的概念，推广了双曲迭代函数系统的概念。基于一个分别使用Plotkin幂域和概率幂域的域理论模型，证明了弱双曲IFS的吸引子的存在唯一性和带概率的弱双曲IFS的不变测度的存在唯一性，推广了Hutchinson关于双曲IFS的经典结果。我们还提出了有限算法来获得吸引子和不变测度的离散和数字化逼近，扩展了双曲ifs的相应算法。然后，我们证明了弱双曲循环IFS的不变量分布的存在唯一性，并给出了在数字屏幕上生成该不变量分布的算法。我们利用广义黎曼积分给出了几乎所有连续函数关于这个分布的期望值的公式。对于双曲循环ifs和Lipschitz图，可以估计到任何精度阈值的积分。
We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domain-theoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.