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Rigidity for infinitely renormalizable area-preserving maps

: Gaidashev, D.; Johnson, T.; Martens, M.


Duke mathematical journal 165 (2016), No.1, pp.129-159
ISSN: 0012-7094
ISSN: 1547-7398
Journal Article
Fraunhofer FCC ()

The period-doubling Cantor sets of strongly dissipative Henon-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 Henon-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.