Absolutely Continuous Invariant Measures for Selectors of Multivalued Maps and Their Applications

多值映射选择器的绝对连续不变测度及其应用

• 批准号: RGPIN-2015-03708
• 负责人: Gora, Pawel
• 金额: $ 1.02万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652175
• 关键词: Absolutely; Continuous; Invariant; Measures; Selectors

A Multi-Valued Map (MVM) is a function whose values are subsets of a space rather than individual elements of the space. The main objective of my research is to study the long term statistical behaviour of dynamical systems defined by a MVM. This approach is the basic method of Ergodic Theory in which I specialize. Ergodic Theory is now an established and active research area (four of fourteen Fields medalists in this century were recognized for their contributions to Ergodic Theory).***An economic example of an MVM is a collection of admissible strategies (selectors). An invariant measure statistically describes the long term average behavior of a chosen strategy (a selector). This means that if we can find a selector with a prescribed invariant measure, then we have also found a strategy to obtain the desired statistical behaviour. ***Besides applications in economics, MVMs can be applied to neurology and physics. Since each neuron communicates with many others, MVMs are useful in describing the dynamics of the brain. We believe that invariant measures, particularly absolutely continuous invariant measures (acim), of such maps describe the steady states of the brain.***My main goal is to describe the class of acims preserved by selectors of a given MVM. I have some preliminary results in dimension one. So far, I have considered MVMs with graphs bounded by piecewise expanding maps. An interesting direction of study in economics is to consider this problem in higher dimensions. I have preliminary results in dimension two, but in general this is a difficult problem. ***A related problem is to characterize the invariant measures for selectors that can be achieved as invariant measures of a position dependent random map based on boundary maps. I plan to establish reasonable conditions under which this representation is possible. This type of result is reminiscent of the holographic principle in physics, which claims that all information about the inside of a black hole can be read from the dynamical behavior on its surface.**

多值映射（MVM）是一个函数，其值是空间的子集，而不是空间的单个元素。 我的研究的主要目标是研究由MVM定义的动力系统的长期统计行为。 这种方法是遍历理论的基本方法，我专门研究。遍历理论现在是一个既定的和活跃的研究领域（在这个世纪的十四个领域奖牌获得者中有四个被公认为他们对遍历理论的贡献）。MVM的一个经济学例子是可接受策略（选择器）的集合。一个不变的测量统计描述了所选策略（选择器）的长期平均行为。这意味着，如果我们能找到一个具有指定不变测度的选择器，那么我们也找到了一个获得所需统计行为的策略。* 除了在经济学中的应用外，MVMs还可以应用于神经学和物理学。由于每个神经元都与许多其他神经元进行通信，因此MVMs在描述大脑动力学方面非常有用。我们相信，这种映射的不变测度，特别是绝对连续不变测度（acim），描述了大脑的稳定状态。我的主要目标是描述由给定MVM的选择器保留的acim类。我在一维空间有了一些初步的结果。到目前为止，我已经考虑了由分段扩展映射限定的图的MVMs。 经济学中一个有趣的研究方向是从更高的维度来考虑这个问题。我在二维空间有了初步的结果，但总的来说，这是一个困难的问题。* 一个相关的问题是描述选择器的不变测度，它可以作为基于边界映射的位置相关随机映射的不变测度来实现。我计划建立合理的条件，使这种代表成为可能。这类结果让人想起物理学中的全息原理，该原理声称，关于黑洞内部的所有信息都可以从其表面的动力学行为中读取。