Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Question

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

A is an integer matrix such that A has an eigenvalue which is a root of unity.

Hint: Use Fourier series to construct a non-constant invariant function.

Not sure how to approach this question. Somehow you can use that there will exists a $\underline{n} \neq \underline{0} \in \mathbb{Z}^d$ with $(A^T)^k \underline{n} = \underline{n} $

Answer 1

Take $B$ to be the complement of the linear space generated by your $n$. Clearly, this has full measure. Consider $f$ to be the characteristic function for $B$. This is an invariant function for $T^k_A$, but not necessarily invariant for just $T_A$. Now, what techinique have you learned that makes functions into invariant functions? Play with it a bit.