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

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

Kneading theory is a combinatorial tool to understand the dynamics of a piecewise monotone map from the interval to itself and was developed by Milnor and Thurston (Lecture Notes 1988).

The set of kneading sequences, denoted as $\mathfrak{K}$, consists of all sequences $\bar{k}^n$ for $n \in \mathbb{Z}$. Each kneading sequence $\bar{k} = \bar{k}^n$, where $n \in \mathbb{Z}$, follows the structure:

$$ \bar{k} = \hspace{.2cm}+^{\hspace{-.5cm}\infty}\hspace{.2cm} w \pm * \overrightarrow{k}_{\hspace{-.1cm} 0},$$

where $\hspace{.2cm}+^{\hspace{-.5cm}\infty}\hspace{.2cm} = \dots + + +$, $w = w_0 \dots w_m$ for some $m \in \mathbb{N}_0$, $\overrightarrow{k}_{\hspace{-.1cm} 0} = k_0 k_1 k_2 \dots$, $w_0 = -$ and $k_0 = +$. The symbols $w_i$ and $k_j$ belong to the set $\{ -, + \}$ for $i = 1, \dots, m$ and $j \in \mathbb{N}$, and the star $*$ denotes the position of the 0th coordinate. The symbol $\pm$ can be replaced by either $+$ or $-$.

Problem 1:Describe the set of kneading sequences $\mathfrak{K}$.

Question 1:Is it true that any two distinct kneading sequences determine a unique pair of parameters $(a,b)$, and in that way govern all the other kneading sequences of $\mathfrak{K}$?

We define Hénon, and Lozi maps ${\widetilde{H}}_{a,b}, L_{a,b} \colon \mathbb{R}^2 \to \mathbb{R}^2$ are defined as $$ {\widetilde{H}}_{a,b}(x,y) = (1 + y - ax^2, bx), \quad L_{a,b}(x,y) = (1 + y - a|x|, bx), $$ respectively. Observe that ${\widetilde{H}}_{a,b}$ is affinely conjugated to $H_{\sqrt{b},-a}$ from the introduction.

Problem 2:Describe the set of kneading sequences $\mathfrak{K}$ of the Hénon map $H_{a,b}$, with $(a,b) \in \mathcal{WY}$.

Question 2:Is it true that any two distinct kneading sequences of the Hénon map $H_{a,b}$, with $(a,b) \in \mathcal{WY}$, determine a unique pair of parameters $(a,b)$, and in that way govern all the other kneading sequences in $\mathfrak{K}$ of $H_{a,b}$?

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