## Institute for Mathematical Sciences

## Preprint ims15-01

** Peter Hazard, Marco Martens and Charles Tresser**
*Infinitely Many Moduli of Stability at the Dissipative Boundary of Chaos*

Abstract: 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 attractors 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>0, the maximal moduli number for non-chaotic but non area-contracting maps in
the interior of the set of zero-entropy is infinite.

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