Real Hénon maps (by P. Berger)

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

Question 1:Can we find a Hénon map (of some degree) which admits a wild attractor, i.e., a Cantor set which attracts a set of Lebesgue positive measure and which is strictly included in a transitive set ?

Refer to Van Strien (2010) who asked whether a suitable analog of wild attractors could exist for Hénon maps (of some degree).

A second question posed concerns stochastic attractors. First, it's noteworthy that ensuring the existence of a parameter $c \in \mathbb{R}$ such that the quadratic map $x \mapsto x^2+c$ exhibits an absolutely continuous measure is straightforward. Simply selecting a parameter $c$ where the post-critical orbit is finite but not periodic suffices. In two dimensions, establishing the existence of a positive measure set of parameters for Hénon maps that exhibit an attractor supporting an invariant SRB measure—a measure whose conditional measures along unstable curves are absolutely continuous—is a significant result, the proof of which remains challenging and lengthy (see [1] for references).

Question 2:Is there a concise proof for the existence of SRB measures for certain parameters of Hénon maps?

A positive response to this question could aid in discovering new instances of stochastic attractors in Hénon maps.

Another question concerns the counterpart of Hofbauer-Keller phenomenon, and stated as follow.

Question 3:Do there exist $\varepsilon>0$ and a locally dense set\footnote{i.e., whose closure has non-empty interior.} $E$ of quadratic H\'enon maps for which every point starting in some set of Lebesgue measure at least $\varepsilon$ has non-statistical behavior?

Question 4:Is there a real, conservative, polynomial Hénon map with an annular rotation domain? Is there an open set of such real, conservative, polynomial Hénon maps?

Recal that a symplectomorphism is an analytic map which extends to a holomorphic map of $\mathbb{C}^2$.

Question 5:Is there an entire symplectomorphism of $\mathbb{R}^2$ with a non trivial annular rotation domain?

Question 5 is related to the following problem

Question 6:Construct an entire symplectic automorphism of $\mathbb{C}^2$ without periodic point and with non-empty set of (Lyapunov) unstable points with bounded orbit.

For further discussions see from P. Berger see [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