Henon-like maps with arbitrary stationary combinatorics

TL;DR

This paper extends the renormalisation framework for Hénon-like maps to those with arbitrary stationary combinatorics, revealing universal structures and non-rigidity in their invariant Cantor sets.

Contribution

It generalizes the renormalisation operator to broader classes of Hénon-like maps and analyzes the properties of their invariant Cantor sets.

Findings

Renormalisation applies to maps with arbitrary stationary combinatorics under strong dissipation.

Invariant Cantor sets are acted upon like p-adic adding machines.

The invariant Cantor set is non-rigid and cannot have a continuous invariant line field.

Abstract

We extend the renormalisation operator introduced in \cite{dCML} from period-doubling H\'enon-like maps to H\'enon-like maps with arbitrary stationary combinatorics. We show the renormalisation picture holds also holds in this case if the maps are taken to be \emph{strongly dissipative}. We study infinitely renormalisable maps $F$ and show they have an invariant Cantor set $\mathcal{O}$ on which $F$ acts like a $p$-adic adding machine for some $p>1$. We then show, as for the period-doubling case in \cite{dCML}, the sequence of renormalisations have a universal form, but the invariant Cantor set $\mathcal{O}$ is non-rigid. We also show $\mathcal{O}$ cannot possess a continuous invariant line field.