Exercise 65, § 1

It is assumed that G is compact, endowed with its normalized Haar measure $\mu$. Let $f: G\rightarrow G$ be a continuous surjective homomorphism.

a) We have $f(\mu) =\mu$.

b) The mapping $f$ is $\mu$-ergodic (cf. exercise 16 of I, p. 190) if and only if there does not exist a character $\chi \not=e_G$ of G and an integer $n\geqslant 1$ such that $\chi \circ f^n=\chi$.

c) Let $d\geqslant 1$ be an integer and let $a$ be a matrix of type $(d, d)$ with integer coefficients and determinant 1. The mapping $f_a:\mathbf{T}^d\rightarrow \mathbf{T}^d$ such that $f_a(x) =ax$ for all $x\in \mathbf{T}^d$ is ergodic with respect to the normalized Haar measure on $\mathbf{T}^d$ if and only if the spectrum of the matrix $a$ in $\mathscr{L}(\mathbf{C}^n)$ contains no root of unity.