CAREER: Unifying approaches to non-uniform hyperbolicity

University Of Houston, Houston TX

Investigators

Abstract

There are many phenomena in the real world that we perceive as having random, unpredictable behavior, despite the fact that they follow underlying rules that are completely deterministic and well understood. For example, when we roll a pair of dice, they fly and bounce according to physical laws that are non-random, but the outcome of the roll is nevertheless random. A more sophisticated example is weather; although we understand atmospheric dynamics and can make reliable short-term predictions, the exact behavior of the weather several weeks from now remains largely random and unpredictable. The theory of hyperbolic dynamical systems studies the mechanism that drives this evolution from predictability to randomness. When this mechanism operates consistently no matter what state the system is in, the system is said to be "uniformly hyperbolic". Such systems are well-understood, but this uniformity condition is so restrictive that it rarely applies to physically realistic examples. A more realistic condition is "non-uniform hyperbolicity", where the increase in randomness only happens some of the time, but nevertheless appears eventually for typical initial conditions. Several different approaches have been used to study non-uniformly hyperbolic systems, each with its own advantages and disadvantages. The goal of this project is to develop a unified theory of non-uniform hyperbolicity by clarifying and strengthening the connections between these existing approaches; by introducing new tools; and by studying new classes of examples. Historically, the two most successful approaches to non-uniform hyperbolicity are Pesin theory (introduced in 1976) and Young towers (introduced in 1998). Recent work by the PI and his co-authors has introduced "effective hyperbolicity" and "non-uniform specification". All four approaches study non-uniform hyperbolicity via invariant measures, but the connections between them are not yet clear, and different approaches have different applications. For example, Pesin theory gives powerful results once an appropriate invariant measure has been selected, but the other three approaches are more useful for finding distinguished invariant measures such as SRB measures and equilibrium states. Similarly, Young towers yield stronger statistical properties than the other approaches, but construction of a tower may be quite difficult. Recently Sarig gave a construction of countable Markov partitions using Pesin theory, which allows a tower to be built for any hyperbolic measure, but gives no information on the rate of decay of the tail of the tower, which is necessary for statistical properties such as the central limit theorem. Preliminary results by the PI and his co-authors give an alternate construction that yields estimates on the decay rate; the PI will develop this approach to obtain strong statistical properties for a broad class of hyperbolic measures, including the SRB measures constructed using effective hyperbolicity and the equilibrium states constructed using non-uniform specification. As a first concrete example, geodesic flow in non-positive curvature will be considered; these techniques are expected to give rapid decay of correlations for the unique measure of maximal entropy and other equilibrium states, and to give polynomial decay of correlations for the regular component of Liouville measure (subject to geometric conditions on the manifold). More generally, the connection between the four approaches will significantly strengthen the tools of effective hyperbolicity and non-uniform specification; it will also make the the powerful theory of Young towers easier to apply, and give a precise sense in which all four approaches are equivalent. A longer-term goal is to extend the class of rigorously understood non-uniformly hyperbolic examples; one important planned application is to Teichmuller flow, which has deep geometrical significance.