Thermodynamics and statistics of non-uniformly hyperbolic dynamical systems

University Of Houston, Houston TX

Investigators

Abstract

It is natural to ask how much our knowledge of the present tells us about the future. If we study some system that evolves in time according to known rules, what predictions can we make based on observing the present state of the system? It is by now well-understood that even relatively simple systems can display chaotic behavior, in which our ability to make exact predictions decays quite quickly as we look further ahead. In this case one may hope to treat the system as a random process and make statistical predictions using tools from probability theory. This approach has been successfully carried out for uniformly hyperbolic systems, the most strongly chaotic. However, most physically realistic examples fall outside of this class, including important models from meteorology (the Lorenz system), population dynamics (the logistic map), and others. This motivates the study of non-uniformly hyperbolic systems, where the present state of knowledge is much less complete. There has been progress towards understanding certain classes of non-uniformly hyperbolic systems, but many open problems remain, both for systems that have been studied and for broader classes of systems. This research project will give new results for several important classes of non-uniformly hyperbolic systems, and is an important step in developing a more complete understanding of physically relevant systems displaying chaotic behavior. Key elements of the uniformly hyperbolic theory include existence and uniqueness results for equilibrium states and SRB measures, together with statistical properties for these measures, such as the central limit theorem governing long-term fluctuations of observations around an expected average, and large deviations principles describing the probability of outcomes far from that average. Some of these results, but not all, are known for classes of one-dimensional maps and their perturbations (strongly dissipative maps), partially hyperbolic systems, and geodesic flows. The PI will extend these results to weakly dissipative maps and more general geodesic flows, and will strengthen existing results in all three categories. The key innovation making these extensions possible is the introduction by the PI and his co-authors of new tools for non-uniform hyperbolicity: a notion of "effective hyperbolicity" for the construction of SRB measures, and a notion of "thermodynamic specification" for uniqueness and statistical properties of equilibrium states. These tools have already yielded a number of new results and have clear applicability to broader classes of systems.