# Example of a Chaotic discrete dynamical system in dimension 2

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.

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 $$$$\label{eq:1} \liminf_{n\to+\infty} d\big(f^n(x),f^n(y)\big) = 0, \quad\forall x,y\in C_f.$$$$

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 $$$$\label{eq:2} \limsup_{n\to+\infty} d(f^n(x),f^n(y))>0, \quad\forall x,y\in\mathbb{T}^2,\ x\neq y.$$$$

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.

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.

• Thank you for the answer. It seem to me like the system is non autonomous right ?
– kiki
Commented Sep 22, 2019 at 6:05