A stationary chain need not be ergodic

The identity matrix $P=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ is a transition matrix, and $\pi$ is invariant: $(\pi P)(0)=\pi (0)\cdot 1=1\mathrm{/}2=\pi (0)$ and likewise at $1$, which is exactly the identity of [F2].

The event $A:=\{z\in E^{{\mathbb{N}}_0}:z_n=0\text{ for infinitely many }n\}={\bigcap }_{m\ge 0}{\bigcup }_{n\ge m}\{z:z_n=0\}$ is a countable Boolean combination of coordinate events and is therefore measurable.

Both states are absorbing, so $X_n=X_0$ for every $n\ge 0$; hence every finite-dimensional law of $X$ is the law of the constant tuple $(X_0,\dots ,X_0)$, which is unchanged by any nonnegative time shift, and the chain is strictly stationary in the sense of [F3]. Its canonical path law is ${\mathbb{P}}_{\pi }=\frac{1}{2}{\delta }_{\bar{0}}+\frac{1}{2}{\delta }_{\bar{1}}$, where $\bar{0}=(0,0,0,\dots )$ and $\bar{1}=(1,1,1,\dots )$.

The event $A$ is strictly shift-invariant: ${\theta }^{-1}A=\{z:\theta z\in A\}=\{z:z_{n+1}=0$ for infinitely many $n\}=A$, because deleting the first coordinate of a sequence does not change whether infinitely many of its entries vanish.

The left shift preserves ${\mathbb{P}}_{\pi }$: by step 2.1 the path law is supported on the two fixed paths $\bar{0},\bar{1}$, and $\theta \bar{0}=\bar{0}$, $\theta \bar{1}=\bar{1}$, so ${\mathbb{P}}_{\pi }({\theta }^{-1}B)={\mathbb{P}}_{\pi }(B)$ for every measurable $B$.

Evaluating at $A$: $\bar{0}\in A$ and $\bar{1}\mathrm{\notin }A$, so ${\mathbb{P}}_{\pi }(A)=\frac{1}{2}$, and by step 2.2 this is the measure of a strictly invariant event; since $\frac{1}{2}\mathrm{\notin }\{0,1\}$, [F4] shows that the canonical shift is not ergodic, even though ${\mathbb{P}}_{\pi }$ is shift-invariant by step 3.1.

Boundary and axiom cases: the event $A$ is strictly invariant, not merely invariant modulo null sets, and step 2.2 verifies the identity on the whole path space; the value $1\mathrm{/}2$ is neither $0$ nor $1$, so the criterion of [F4] genuinely fails; $A^c$ and the events "infinitely many ones" behave the same way; if $\pi$ were concentrated on $0$ or on $1$ the chain would be ergodic, so the mixture is essential; the chain is reducible with two communicating classes $\{0\}$ and $\{1\}$, which is exactly why the ergodicity result for irreducible chains does not apply; the explicit description of ${\mathbb{P}}_{\pi }$ in step 2.1 makes no selection and no choice principle is used; and no convergence claim is made, the example refuting only the implication "stationary $\Rightarrow$ ergodic".