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

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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.**