Arc exchange systems and renormalization

We describe the construction of stable arc exchange systems from the stable laminations of hyperbolic diffeomorphisms. A one-to-one correspondence is established between (i) Lipshitz conjugacy classes of C^1+H stable arc exchange systems that are C^1+H fixed points of renormalization and (ii) Lipshitz conjugacy classes of C^1+H diffeomorphisms f with hyperbolic basic sets Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. Let HD^s(Λ) and HD^u(Λ) be, respectively, the Hausdorff dimension of the stable and unstable leaves intersected with the hyperbolic basic set Λ. If HD^u(Λ)=1, then the Lipschitz conjugacy is in fact a C^1+H conjugacy in (i) and (ii). We prove that if the stable arc exchange system is a $C^{1+HD^{s}+\alpha}$ fixed point of renormalization with bounded geometry, then the stable arc exchange system is smoothly conjugate to an affine stable arc exchange system.