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

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.