Infinitely Many Moduli of Stability at the Dissipative Boundary of Chaos

TL;DR

This paper demonstrates that within area-contracting Hénon-like maps with zero topological entropy, there are infinitely many moduli of stability, indicating a rich diversity of topological types that cannot be captured by finitely many parameters.

Contribution

It proves the existence of infinitely many moduli of stability for certain non-chaotic Hénon-like maps, extending understanding of their topological complexity.

Findings

Infinitely many moduli of stability exist for non-chaotic area-contracting Hénon-like maps.

The number of parameters needed to classify topological types increases with the number of periods.

Non-area-contracting zero-entropy maps can have infinitely many moduli, contrasting with the area-contracting case.

Abstract

In the family of area-contracting H\'enon-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 H\'enon-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 H\'enon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in $\mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting H\'enon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting H\'enon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $\{1,\, 2,\dots,\,2^{n-1}\}$ 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 $k\geq 1$, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.