A list of open problems on Hénon Maps

Problem's Description

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].

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

1. Open Real Hénon maps (by P. Berger)

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

3. Open Symbolic dynamics for real Hénon and Lozi maps (By S. \v Stimac)