Example of a Chaotic discrete dynamical system in dimension 2

Question

I am looking for examples of discrete dynamical systems in dimension 2 that are :

1) Chaotic dynamical system in Devaney's sense in dimension 2 ? 2) Chaotic dynamical system in Li-Yorke sense but not chaotic in Devaney's sense.

Thank you.

Answer 1

Kiki, regarding question 1), any Anosov diffeomorphism on $\mathbb{T}^2$ is Devaney chaotic.

Regarding question 2), the are minimal (i.e. every orbit is dense) Li-Yorke chaotic diffeomorphisms on $\mathbb{T}^2$, and since they are periodic point free, they cannot be Devaney chaotic.

The existence of such minimal diffeomorphisms can be proven as follows.

In a joint work with A. Koropecki https://arxiv.org/abs/0902.2474 we proposed the following definition: A homeomorphism $f\colon\mathbb{T}^2\to\mathbb{T}^2$ is said to be {weak spreading if for a lift $\hat f\colon\mathbb{R^2}\to\mathbb{R}^2$ of $f$, for any non-empty open set $U\subset\mathbb{R}^2$, any $\varepsilon>0$ and any $R>0$, there is an integer number $n>0$ such that $\hat{f}^{-n}(U)$ is $\varepsilon$-dense in a ball of radius $R$ (in $\mathbb{R}^2$).

By classical Baire's arguments one can show that for every weak spreading homeomorphism $f\colon\mathbb{T}^2\to\mathbb{T}^2$ there exists a Baire generic set $C_f\subset\mathbb{T}^2$ such that \begin{equation} \label{eq:1} \liminf_{n\to+\infty} d\big(f^n(x),f^n(y)\big) = 0, \quad\forall x,y\in C_f. \end{equation}

Now cosnsider the set $\mathcal{O}^\infty(\mathbb{T}^2) = \{h\circ T_\alpha\circ h^{-1} : \alpha\in\mathbb{T}^2,\ h\in\mathrm{Diff}^\infty(\mathbb{T}^2)\}$, where $T_\alpha\colon x\mapsto x+\alpha$ denotes the rigid translation and let $\overline{\mathcal{O}^\infty(\mathbb{T}^2)}$ denote its $C^\infty$-closure.

A. Fathi and M. Herman showed that there is a Baire generic set $\mathscr{C}_0\subset\overline{\mathcal{O}^\infty(\mathbb{T}^2)}$ such that every diffeomorphism of $\mathscr{C}_0$ is minimal. (see https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=fathi%20AND%20HErman&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=482843)

In https://arxiv.org/abs/0902.2474 we showed that there is a generic set $\mathscr{C}_1\subset\overline{\mathcal{O}^\infty(\mathbb{T}^2)}$ such that every element of $\mathscr{C}_1$ is weak spreading, and hence satisfies the above liminf condition.

Finally, it is known (sorry, I could not find a reference for this) that there is a generic set $\mathscr{C}_2\subset\overline{\mathcal{O}^\infty(\mathbb{T}^2)}$ such that every element of $\mathscr{C}_2$ is rigid, i.e. if $f\in\mathscr{C}_2$, then there is a sequence $n_j\to+\infty$ such that $f^{n_j}\to id$ in the $C^0$ topology, when $n_j\to+\infty$. On the other hand, it is clear that, if $f$ is rigid, then \begin{equation} \label{eq:2} \limsup_{n\to+\infty} d(f^n(x),f^n(y))>0, \quad\forall x,y\in\mathbb{T}^2,\ x\neq y. \end{equation}

So, puting all these properties together one can see that every element of $\mathscr{C}_0\cap\mathscr{C}_1\cap\mathscr{C}_2$ is minimal and Li-York chaotic.

Answer 2

For another answer to Question 1, consider Julia sets of holomorphic functions of one complex variable (e.g., polynomials or rational maps of degree $\geq 2$, or transcendental entire functions). Any such map is both Devaney and Li-Yorke chaotic on its Julia set.

A famous result of Misiurewicz shows that $J(\exp)=\mathbb{C}$ is a famous result of Misiurewicz. In particular, the complex exponential map $\mathbb{C}\to\mathbb{C}$ is chaotic on the complex plane. (Compare my Monthly paper with Shen, The exponential map is chaotic, arxiv:1408.1129.)

Similarly, for the rational map $z\mapsto (z^2+1)/(4z(z^2-1))$, the Julia set is the entire Riemann sphere. (See Beardon, Iteration of rational functions, Section 4.3.) So this is an analytic system on the sphere which is Devaney chaotic.

Answer 3

1) Chaos in the sense of Devaney for two-dimensional time-varying generalized symbolic dynamical systems