Are product measures the only $T$-invariant measures if $T$ is the left shift on $\{0,1,\ldots,k-1\}^\mathbb Z$?

Question

Consider $X=\{0,1,\ldots,k-1\}^\mathbb Z$ equipped with the $\sigma$-algebra $\mathcal C$ generated by the cylinder sets $$\{x\in X:x_m=i_m,\ldots,x_n=i_n\}\tag{$m\leq n$}.$$ It is well known that the $p$-Bernoulli measure $\mu$, defined by letting $$\mu\{x\in X:x_m=i_m,\ldots,x_n=i_n\}=p_{i_m}\cdots p_{i_n}$$ is $T$-invariant (and ergodic) for the left shift $T:X\to X:x\mapsto y$, where $y_n=x_{n+1}$. However, I was wondering if there are any other $T$-invariant (not necessarily ergodic) measures. If I understand it correctly, this post seems to imply that no other $T$-invariant measure exists, but I don't see why such another measure is impossible.

Answer 1

There are many, many other shift-invariant measures. One very easy example is something like $\mu = \frac12 \delta_{x} + \frac12 \delta_{Tx}$, where $x$ is the periodic sequence $010101\cdots$.

Another natural example is the Markov measure generated by transition probabilities $(P_{ij})_{0 \leq i,j \leq k-1}$ and the corresponding stationary distribution $(\pi(0), \dots, \pi(k-1))$. The Markov measure $\mu$ is defined on cylinders by $$ \mu\{x \in X : x_m \cdots x_{n} = i_{m} \dots i_n\} = \pi(i_m) P_{i_m i_{m+1}} \cdots P_{i_{n-1} i_n}. $$ This example is called a Markov measure because it's just the dynamical systems way of talking about the joint distribution for sample paths of a finite state Markov chain.

Also, you can check that the first example is ergodic, and it's a classical theorem that the Markov measure is ergodic if the Markov chain is irreducible.

There are plenty of other examples besides these. For shift systems, the set of invariant measures and the set of ergodic invariant measures are both huge.