Infinitely many moduli of stability at the dissipative boundary of chaos

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 attractor 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 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.