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

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。我研究的主要目的是研究由MVM定义的动力系统的长期统计行为。这种方法是我所擅长的遍历理论的基本方法。遍历理论现在是一个成熟且活跃的研究领域（本世纪14位菲尔兹奖得主中有4位因对遍历理论的贡献而得到认可）。MVM的一个经济例子是可接受策略（选择器）的集合。一个不变的度量在统计学上描述了所选策略的长期平均行为（选择器）。这意味着，如果我们能够找到具有规定不变度量的选择器，那么我们也就找到了获得所需统计行为的策略。除了在经济学中的应用，MVMs还可以应用于神经学和物理学。由于每个神经元与许多其他神经元通信，mvm在描述大脑动态方面很有用。我们相信，这些图的不变测度，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的选择器保存的acims类。我在维一上有一些初步的结果。到目前为止，我已经考虑了带有分段扩展图的图的mvm。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。我在二维中得到了初步结果，但通常这是一个难题。***一个相关的问题是描述选择器的不变测度，这些选择器可以作为基于边界图的位置相关随机映射的不变测度。我计划建立合理的条件，在此条件下，这种表现是可能的。这种类型的结果让人想起物理学中的全息原理，它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

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.**