Invariant manifolds for non-differentiable operators

Hyperbolicity of renormalization for dissipative gap mappings

“…To conclude the proof of Theorem A, we make use of the graph transform. We refer the reader to Section 2 of [25], for the proofs of some of the results in this section. Let X 0 = {w ∈ C(B, [0, 1]) : for all p, q ∈ graph(w), q − p / ∈ C r,δ }.…”

“…As a result, the renormalization operator for gap mappings does not seem to possess a natural extension to the limits of renormalization (c.f. [25]).…”

“…For Lorenz mappings of certain monotone combinatorial types, [30] proved that there exists a global two-dimensional strong unstable manifold at every point in the limit set of renormalization using this approach. In [25] they studied renormalization acting on the decomposition space for infinitely renormalizable critical circle mappings with a flat interval. They proved that for certain mappings with stationary, Fibonacci, combinatorics that the renormalization operator is hyperbolic, and that the class of mappings with Fibonacci combinatorics is a C 1 manifold.…”

“…In our setting we are able to obtain fairly complete results without any restrictions on the combinatorics of the mappings. In Section 5 we use the estimates of Section 4 and ideas from [25] to show that the renormalization operator is hyperbolic and that the conjugacy classes of dissipative gap mappings are C 1 manifolds.…”

“…[25, Lemma 8.20] Let ϕ ∈ Diff 3 + ([0, 1]). The zoom curve Z : [0, 1] 2 ∋ (a, b) → Z [a,b] ϕ ∈ Diff 2 + ([0, 1]) is differentiable with partial derivatives given by(4.53) ∂Z [a,b] ϕ ∂a = (b − a)(1 − x)Dη((b − a)x + a) − η((b − a)x + a) ∂Z [a,b] ϕ ∂b = (b − a)xDη((b − a)x + a) + η((b − a)x + a).…”