The average R(t) = R φ dμt of a smooth function φ with respect to the SRB measure μt of a smooth one-parameter family ft of piecewise expanding interval maps is not always Lipschitz [4], [18]. We… (More)

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and “complex bounds”,… (More)

We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus.… (More)

In the space of C piecewise expanding unimodal maps, k ≥ 2, we characterize the C smooth families of maps where the topological dynamics does not change (the “smooth deformations”) as the families… (More)

We give two new proofs that the Sinai–Ruelle–Bowen (SRB) measure t 7→ μt of a C2 path ft of unimodal piecewise expanding C3 maps is differentiable at 0 if ft is tangent to the topological class of… (More)

Consider deterministic random walks F : I × Z → I × Z, defined by F (x, n) = (f(x), ψ(x) + n), where f is an expanding Markov map on the interval I and ψ : I → Z. We study the universality… (More)

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… (More)

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the… (More)

We consider a piecewise analytic expanding map f : [0, 1] → [0, 1] of degree d which preserves orientation, and an analytic positive potential g : [0, 1] → R. We address the analysis of the following… (More)

We study the dynamics of the renormalization operator for multimodal maps. In particular, we develop a combinatorial theory for certain kind of multimodal maps. We also prove that renormalizations of… (More)