Renormalization theory for multimodal maps

We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.Comment: 37 pages, 4 figure

Corrigendum to "Linear response formula for piecewise expanding unimodal maps," Nonlinearity, 21 (2008) 677-711

The claim in Theorem 7.1 for dense postscritical orbits is that there exists a sequence tn (not for all sequences).Comment: Latex, 2 pages, to appear Nonlinearit

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

We consider C^2 families of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of f_t depends differentiably on t, as a distribution of order 1. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of the acim for a Benedicks-Carleson map f_t, in terms of a single smooth function and the inverse branches of f_t along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v(x)=\alpha (f (x)) - f'(x) \alpha (x) has a continuous solution \alpha, if f is Benedicks-Carleson and v is horizontal for f.Comment: Typos corrected. Banach spaces (Prop 4.10, Prop 4.11, Lem 4.12, Appendix B, Section 6) cleaned up: H^1_1 Sobolev space replaces C^1 and BV, L^1 replaces C^0, and H^2_1 replaces C^2. Details added (e.g. Remark 4.9). The map f_0 is now C^4. 61 page