INFINITELY MANY MODULI OF STABILITY AT THE DISSIPATIVE BOUNDARY OF CHAOS

In the family of area-contracting Henon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Henon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Henon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in R-2. Our proof uses recent results about infinitely renormalisable area-contracting Henon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Henon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to {1, 2,..., 2(n-1)} or an initial segment of this n-tuple) increases with the number of periods. In comparison, among C-k-embeddings of the 2-disk with k >= 1, the maximal moduli number for non-chaotic but non-area-contracting maps in the interior of the set of zero-entropy is infinite.