Validated numerics for Iterated Function Schemes, Dynamical Systems and Random Walks

迭代函数方案、动力系统和随机游走的经过验证的数值

基本信息

批准号：
EP/W033917/1
负责人：
Mark Pollicott
金额：
$ 51.62万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW033917%2F1
关键词：
Validated numerics Iterated Function Schemes

项目摘要

In many areas of mathematics its is useful to have estimates for numerical values. In mathematical analysis this may be the notion of dimension which describes the size of sets. In the context of Ergodic Theory and dynamical systems this includes, for example, the Lyapunov exponents which measure how typical nearby orbits separate as the system evolves. In the setting of random walks on hyperbolic groups (generalizing the famous "drunkard's walk" in one dimension) it is the dimension of an associated measure (which measures how "spread out" the measure is). Whereas these values give qualitative information in each of these settings, there are particularly interesting applications when we require a precise knowledge of their values. That is, we need to know their values really do satisfy some inequality and this has two ingredients. Firstly, having a method to approximate the number which is efficient and accurate. Secondly, this result is validated - to the extent that we can have complete confidence in these results that comes from the underpinning abstract mathematics. Here the emphasis is less on the problem of computation and more on the development of an efficient algorithm and making the connection with the applications.The use of explicit numerical estimates and their surprising applications other areas of mathematics is illustrated by the density one Zaremba theorem of Fields medallist Bourgain and Kontorovich in number theory. By the Euclidan algorithm it is known that any rational number p/q can be written as a finite continued fraction, i.e., there exist natural numbers a_1, ..., a_n with p/q = 1/(a_1+1/(a_2+...)). Bourgain and Kontorovich showed that for typical q there exists a p and a_1, ..., a_n taking one of the values 1,2,3,4 or 5, with p/q = 1/(a_1+1/(a_2+...)). This crucially depends on a certain associated Cantor set in the unit interval having dimension greater than 5/6.

在许多数学领域，对数值的估计很有用。在数学分析中，这可能是描述集合大小的维度的概念。在沿着沿阵行的理论和动力学系统的背景下，这包括例如，lyapunov指数，这些指数衡量附近的典型轨道随着系统的发展而分离。在双曲线群体的随机步行设置（将著名的“醉汉步行”一维概括）中，这是相关措施的维度（衡量了该措施的“分布”是如何）。尽管这些值在这些设置中的每一个中都提供了定性信息，但当我们需要精确了解其价值时，就会有特别有趣的应用程序。也就是说，我们需要知道它们的价值确实满足了一些不平等，这有两种成分。首先，使用一种方法来近似高效且准确的数字。其次，该结果得到了验证 - 在一定程度上，我们可以完全相信来自基础的抽象数学的结果。在这里，重点不太在于计算问题，而更多地是在有效算法的开发和与应用程序建立联系上。使用显式数值估计及其令人惊讶的应用程序其他数学领域的其他领域则用密度为Zaremba Theorem of Mentive of Zaremba定理田野奖牌获得者在数字理论中。通过Euclidan算法，众所周知，任何有理数P/Q都可以写入有限的持续分数，即存在自然数A_1，...，A_N，a_n，p/q = 1/（A_1+1/（A_2+） ...））。 Bourgain和Kontorovich表明，对于典型的Q，存在p和a_1，...，a_n，以1,2,3,4或5的值之一，p/q = 1/（a_1+1/（a_2+..））。这至关重要的是在尺寸大于5/6的单位间隔中设置的某个相关的cantor设置。