Dissipative real Hénon maps (by S. Crovisier and E. Pujals)

General Description View the group

In April 2023, a workshop titled "Exploring Hénon Maps: Real, Complex, and Beyond" took place at BIRS, Banff, with the goal of uniting global specialists focusing on various aspects of Hénon maps. See [1] for a detail description and all problems announced during the meeting.

The Hénon map, also referred to as the Hénon–Pomeau attractor/map, is a discrete-time dynamical system that is extensively studied due to its chaotic behavior. The map transforms a point $(x_n, y_n)$ in the plane to a new point defined by the equations:

$$ \begin{cases} x_{n+1}=1-ax_{n}^{2}+y_{n}\\ y_{n+1}=bx_{n} \end{cases} $$

The behavior of the map depends on two parameters, $a$ and $b$, with classical values of $a = 1.4$ and $b = 0.3$ resulting in chaotic dynamics. However, for different parameter values, the map may exhibit chaotic, intermittent, or periodic behavior. The Hénon map was initially introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, initial points in the plane tend to either approach a set of points known as the Hénon strange attractor or diverge to infinity. This attractor is a fractal, exhibiting smoothness in one direction and a Cantor set structure in another. Numerical estimates suggest a correlation dimension of $1.21 \pm 0.01$ or $1.25 \pm 0.02$ (depending on the dimension of the embedding space) and a Box Counting dimension of $1.261 \pm 0.003$ for the attractor of the classical map.

This group of problem contains several problems from this event, that appeared in [1].

Problem's Description

Consider the quadratic real Hénon maps $$f_{c, b} (x, y):= (x^2+c-by, x).$$

The map $f_{c,b}$ is called mildly dissipative if $|b|< 1$ and if for any ergodic invariant measure $\mu$, not supported on a sink, and for $\mu$-almost every point $x$, both components of the stable curve $W^s(x)\setminus \{x\}$ are unbounded.

Question 1:Which real Hénon maps $f_{c,b}$ are mildly dissipative? Is this property satisfied by all Hénon maps with $|b|< 1$?

Consider the locus in the parameter space where the topological entropy vanishes $$\mathcal{E}_0:=\{(b,c)\in \mathbb{R}^2, h_{\text{top}}(f_{c,b})=0\}.$$

Question 2:Consider any infinitely renormalizable mildly dissipative map $f_{c,b}\in\mathcal{E}_0$. Is it the limit of maps with positive entropy? Is it the limit of Morse-Smale maps?

Question 3:Let us consider any infinitely renormalizable mildly dissipative map in $\mathcal{E}_0$. Does the sequence of renormalizations converge?

Question 4:Does there exist a mildly dissipative map $f_{c,b}\in \mathcal{E}_0$ which admits arbitrarily deep renormalizations with odd periods?

Question 5:For any real Hénon map, does the closure of the set of periodic orbits support all the invariant probability measures?

Question 6:For any map $f_{c,b}$ and any infinite set of periodic saddles $(p_n)$ which are pairwise not homoclinically related, do we have $\min(|\lambda^-(p_n)|,|\lambda^+(p_n)|)\underset{n\to \infty}\longrightarrow 0$?

1. OtherIs an originHénon maps: a list of open problems

   • Eric Bedford,
   • Pierre Berger,
   • Fabrizio Bianchi,
   • Xavier Buff,
   • Sylvain Crovisier,
   • Tien-Cuong Dinh,
   • Romain Dujardin,
   • Charles Favre,
   • Tanya Firsova,
   • Patrick Ingram,
   • Yutaka Ishii,
   • Liviana Palmisano,
   • Enrique Pujals,
   • Jasmin Raissy,
   • Gabriel Vigny,
   • Sonja Štimac

   arXiv