Microdynamics for Nash maps

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August 2010, 27(3): 1007-1024. doi: 10.3934/dcds.2010.27.1007

Microdynamics for Nash maps

William Geller ^1, , Bruce Kitchens ^1, and Michał Misiurewicz ^1,

1.	Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, United States

Received January 2009 Revised February 2010 Published March 2010

Abstract
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We investigate a family of maps that arises from a model in economics and game theory. It has some features similar to renormalization and some similar to intermittency. In a one-parameter family of maps in dimension 2, when the parameter goes to 0, the maps converge to the identity. Nevertheless, after a linear rescaling of both space and time, we get maps with attracting invariant closed curves. As the parameter goes to 0, those curves converge in a strong sense to a certain circle. We call those phenomena microdynamics. The model can be also understood as a family of discrete time approximations to a Brown-von Neumann differential equation.

Keywords: Scaling, Invariant curve..

Mathematics Subject Classification: 37C70, 37E3.

Citation: William Geller, Bruce Kitchens, Michał Misiurewicz. Microdynamics for Nash maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1007-1024. doi: 10.3934/dcds.2010.27.1007

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