19 - METRIC STRUCTURE OF HYPERBOLIC SETS

Summary

In the previous chapter we concentrated on the behavior of purely topological invariants associated with hyperbolic sets. The topological dynamics of a hyperbolic set is closely related to that of a topological Markov chain and since a hyperbolic set appears as an invariant set of a smooth dynamical system it is relevant to study how this topological dynamics is embedded in a smooth manifold. The main conclusion is that all principal structures associated with the dynamics are Hölder continuous and sometimes possess a moderate degree of differentiability (for example, C¹). Higher differentiability is very exceptional. It turns out that Hölder regularity is also natural for treating cohomological equations of the kind discussed in Section 2.9 over hyperbolic dynamical systems. Our main conclusion, the Livschitz Theorem 19.2.1, asserts that periodic obstructions provide complete systems of invariants of Hölder cocycles up to Holder coboundaries. This result as well as its C¹ version has a number of useful applications.

Hölder structures

a. The invariant class of Hölder-continuous functions. Earlier (Section 1.9a and Exercises 1.9.1–1.9.3) we encountered the class of Hölder-continuous functions on the phase space of a dynamical system. It arose there naturally since the space was a sequence space and there was a one-parameter family of naturally defined metrics. We observed that these metrics not only induced the same topology, but had the same class of Hölder-continuous functions.

In this section we will see that the class of Hölder-continuous functions on a hyperbolic set also arises rather naturally. One of the main points of studying such functions is that they will enable us to study the ergodic theory of smooth hyperbolic systems in greater detail.