Functional Norms for Young Towers

We introduce functional norms for hyperbolic Young towers which allow us to directly study the transfer operator on the full tower. By eliminating the need for secondary expanding towers commonly employed in this context, this approach simplifies and expands the analysis of this class of Markov extensions and the underlying systems for which they are constructed. As an example, we prove large deviation estimates with a uniform rate function for a large class of noninvariant measures and show how to translate these to the underlying system. Young towers were introduced in [Y1] as a unified framework in which to view the statistical properties of both uniformly and nonuniformly hyperbolic dynamical systems. Briefly, given a dynamical system f : M , a Young tower is a type of Markov extension F : ∆ with the following representation. Given a reference measure μ, one chooses a reference set Λ ⊂ M of positive measure with a hyperbolic product structure and constructs a return time function R : Λ → Z with certain (Markov) properties. The Young tower is an extension of ∪`≥0f Λ where the ` level of the tower corresponds to those x ∈ f Λ for which R(x) > `. In essence, the Young tower represents Λ as a horseshoe with countably many branches and variable return times. The rate of decay in the measure of the levels of the tower, μ(R(x) > `), gives information about the statistical properties of f ; for example, it reflects the exponential or polynomial decay of correlations (see [CY] for a survey). Young towers have been constructed for many systems: billiards with convex scatterers, including those subject to external forces [Y1, C3, C2], piecewise hyperbolic attractors [Y1, C1], Hénon maps [Y1, BY], nonuniformly expanding maps in one dimension [Y1, WY], and Lorentz attractors [HM]. The strategy in all these papers is the same. One first constructs a Young tower F : ∆ satisfying certain properties (see (P1)-(P5) of Section 1.1). One then defines a quotient tower ∆ = ∆/∼ where x ∼ y whenever x and y lie on the same stable leaf. There are thus ∗Department of Mathematics and Computer Science, Fairfield University, Fairfield CT 06824, USA. Email: [email protected]. The author would like to thank L.-S. Young for helpful discussions, C. Liverani for introducing him to the functional approach, and MSRI, Berkeley, where this work was begun. This research is partially supported by NSF grant DMS-0801139.