Rigidity for infinitely renormalizable area-preserving maps

Abstract

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. 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, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.Comment: 55 pages, incl. references; 2 figure

On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps

“…for w ∈ {0, 1} n and v ∈ {0, 1}, see Lemma 3.3 of [11]. This gives us the following schematic picture of the renormalization microscope.…”

“…We consider two slightly different renormalization schemes: the one introduced by Eckmann et al [5] and the one used by Gaidashev et al [11]. Both schemes are defined for exact symplectic diffeomorphisms of subsets of R 2 .…”

“…In [11] it is also required that F(0, 0) = (0, 0). For such maps it is possible to find a generating function S = S(x, X ) such that…”

“…This explains previously observed universality phenomena in families of such maps. Further investigations of this renormalization have been done by Gaidashev and Johnson [8][9][10] and by Gaidashev et al [11]. In these papers they prove existence of period doubling invariant Cantor sets for all infinitely renormalizable maps and also show that they are rigid.…”

“…Since all area-preserving maps have the same average Jacobian, one would expect, if indeed it is such a classifying invariant, that these Cantor sets are rigid. In [11] a conjecture is made that the average Jacobian is indeed such a classifying invariant.…”