Renormalisation Gives All Surface Anosov Diffeomorphisms with a Smooth Invariant

We prove that there is a natural one-to-one correspondence between xed points of the renormalisation transformation on C 1+ conjugacy classes of C 1+ diieomorphisms of the circle and Anosov diieomorphisms of surfaces with an invariant measure that is absolutely continuous with respect to two-dimensional Lebesgue measure. An important idea in the proof of this is that these conjugacy classes are precisely those for which the induced invariant aane structure on the stable foliation is dual to that of the unstable foliation in the sense that we introduce. We also prove a theorem which gives a new ratio decomposition for the SRB measures of surface Anosov systems. Contents 1. Introduction. 1 2. Smooth train-tracks for Anosov diieomorphisms. 5 3. Renormalisation xed points and train-tracks. 8 4. Duality. 9 5. Basic holonomies. 10 6. AAne structure on the unstable lamination. 11 7. The SRB measures and their ratio decomposition. 14 8. The dual aane structure on the stable lamination. 18 9. The absolute continuity of the 2-dimensional SRB measure. 20 10. Absolute continuity implies duality of the aane structures. 21 References 22 Appendix A. Some commonly used symbols and deenitions. 23