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Differentiable structures on fractal like sets, determined by intrinsic scaling functions on dual Cantor sets. (English) Zbl 0707.58038
Nonlinear evolution and chaotic phenomena, Proc. NATO ASI, Noto/Italy 1987, NATO ASI Ser., Ser. B 176, 101-110 (1988).
[For the entire collection see Zbl 0706.00019.]
This major contribution towards solving the mystery of Feigenbaum universality presents essential ideas in a splendid way, leaving details to the reader.
Consider an embedding of the Cantor set \(C=\{0,1\}^ N\) into \({\mathbb{R}}\) obtained by starting with intervals \(I_ 0\), \(I_ 1\) and for each word w of zeros and ones dividing \(I_ w\) into two subintervals \(I_{w0},I_{w1}\) and a gap \(G_ w\). The numbers length \(I_{w0}/\) length \(I_ w\), length \(mI_{w1}/length I_ w\) and length \(G_ w/\) length \(I_ w\) are said to constitute the ratio geometry of C. They are assumed to be bounded away from zero. Two ratio geometries are considered to be equal if their difference decreases exponentially in length w. The concept of ratio geometry is equivalent to that of a differentiable structure on C - a set of (local) embeddings from C into \({\mathbb{R}}\) with transition functions in \(C^{1+\epsilon}\) for some \(\epsilon >0\). Moreover, the shift map \(J(i_ 0i_ 1i_ 2...)=i_ 1i_ 2..\). on C is in \(C^{1+\epsilon}\) for some \(\epsilon\) if and only if the ratio geometry has a limiting “scaling function” on the dual Cantor set of all left-infinite 0-1 words which is Hölder continuous.
This theory is applied to Feigenbaum-type attractors C of quadratic-like mappings f. Existence of a scaling function for C is equivalent to convergence of the renormalizations of f. Two functions with the same renormalization limit have \(C^{1+\epsilon}\) conjugate attractors.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37E99 Low-dimensional dynamical systems
54H20 Topological dynamics (MSC2010)
37B99 Topological dynamics
37G99 Local and nonlocal bifurcation theory for dynamical systems