Zero entropy Hénon-like maps depend on infinitely many parameters

In the family of area-contracting Hénon-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 Hénon-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 Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractors in R2. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to {1, 2, . . . , 2n−1} or an initial segment of this n-tuple) increases with the number of periods. In comparison, among Ck-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.