Example of Topologically Mixing map on $k$-dimensional cube

Question

Let $k,M$ be positive integers. Is there a simply explicit example of a topologically mixing map on:

• The "cube" $[-1,1]^k$?
• The "disc" $\{x \in \mathbb{R}^k: \|x\|\leq 1\}$?

And what are the points therein with dense, periodic orbits...

Since the product of topologically transitive maps need not be topologically transitive, I cannot build an example from the $1$-dimensional case.

Answer 1

Do you know the tent map $T\colon[0,1]\to[0,1]$, look here? It's a piecewise linear map, with $T(0)=0=T(1)$ and $T(\frac{1}{2})=1$. It has this property, that if you iterate it, its graph will look like many peaks, and for any open subset $U\subset [0,1]$ there is $N\in\mathbb{N}$ such that for any $n>N$ we have $T^n(U)=[0,1]$. It's much stronger then topological mixing and if you take $\Pi^k T\colon [0,1]^k\to[0,1]^k$ it will also have this property, so you should be able to do the cube.

Answer 2

Expanding example on the torus: the doubling map.

A simple mixing example for the torus, $\mathbb T^k = [0,1]^k/\sim$, is simply the doubling map in $k$ dimensions:

$$f:(x_1,x_2,\ldots,x_k) \mapsto (2 x_1,2 x_2, \ldots 2x_k) \mod 1.$$

I.e., $(x_1,x_2,\ldots,x_k) \mapsto \left(D(x_1), D(x_2), \ldots D(x_k)\right)$, where $D:[0,1]\to [0,1]$ is the one dimensional doubling map, passed to the quotient:

$$D(x) = \begin{cases} 2x & 0 \le x \le \frac12, \\ 2x - 1 & \frac12 < x\le 1.\end{cases}$$

Heuristic: why it mixes.

This $f$ mixes for the same reasons as $D$ in one dimension. Namely, given an interval $I$ of length $l>0$ in $[0,1]$, whenever $ n > - \log_2 (l)$, we have $f^n(I)=[0,1]$.

This argument can be generalised to $k$ dimensions as follows:

Every open set contains a small $k$-cube. All of the dimensions of this cube are being enlarged by a factor of $2$ with each iteration of the map, and so with enough iterates will engulf the whole cube $Q$.

That is, for all open sets $U$, for large enough $n$ (depending on the minimum length of $U$); $f^n(U) = \mathbb T$, and in particular meets every other open set $V$ in the cube.

This example generalises to: Take any matrix $A \in \text{GL}_k(\mathbb Z)$ with integer entries, such that all eigenvalues are strictly outside the unit disk ($|\lambda|> 1$ for all eigenvalues $\lambda$).

Then $x\mapsto A x \mod 1$ has similar properties to the doubling map, above.

The harder generalisation of @erfink is that of an Anosov map, where you insist that no eigenvalues of $A$ lie on the unit circle (we may also require $\det(A)=1$), examples being

$$\left( \begin{array}{cc} 2 & 1 \\ 1 & 1 \\ \end{array} \right), \qquad \left( \begin{array}{ccc} 3 & 2 & 1 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \\ \end{array} \right),$$ and so on.

Answer 3

I would like to clarify the definition. I checked two standard dynamical systems references:

Brin, Michael; Stuck, Garrett, Introduction to dynamical systems, Cambridge: Cambridge University Press (ISBN 978-1-107-53894-8/pbk; 978-0-511-75531-6/ebook). xii, 240 p. (2015). ZBL1319.37001.

Katok, Anatole; Hasselblatt, Boris, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and Its Applications. 54. Cambridge: Cambridge Univ. Press. xviii, 802 p. (1995). ZBL0878.58020.

as well as the this Scholarpeda article on topological dynamics by Kolyada and Snoha.

In all of these sources, a topologically mixing dynamical system is required to be continuous. The accepted answer is obviously discontinuous, so it is not topologically mixing by the standard definitions.

I am unaware of any sources which would allow discontinuous topological dynamical systems.