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Rigidity for infinitely renormalizable area-preserving maps. (English) Zbl 1353.37088

From the abstract: “The period-doubling Cantor sets of strongly dissipative Hénon-like maps with different average Jacobian are not smoothly conjugated, as was shown previously. The Jacobian rigidity conjecture says that the period-doubling Cantor sets of two-dimensional Hénon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, for example, the one-dimensional case. The other extreme case is when the maps preserve area, for example, when the average Jacobian is one. Indeed, the main result presented here is that the period-doubling Cantor sets of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.”
Near the renormalization fixed point, each map in the stable manifold of the renormalization operator can be smoothly conjugated to a map in the strong stable manifold, a submanifold of codimension one which has a faster rate of contraction. This is key to the authors’ proof, which relies on the contraction rate being faster than some distortion estimate. The required numerical estimates were carried out in a prior work of the first two authors [Discrete Contin. Dyn. Syst. 36, No. 7, 3651–3675 (2016; Zbl 1354.37046)]. The required estimates appear to hold fortuitously. The authors prove smooth conjugacy in the strong stable manifold and then extend to the general stable manifold. The paper is well written and accessible.

MSC:

37E20 Universality and renormalization of dynamical systems
37F25 Renormalization of holomorphic dynamical systems

Citations:

Zbl 1354.37046
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References:

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