Question

1. Examples of measurable transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).

2. Example of a dynamical system where the following inequality is strict:

   $\sup_{m \in E(T)} h_m(T) < \sup_{\mu \in M(T)} h_\mu(T)$.

Background

Consider $T(x) = x + 1$ over the set of integers $\mathbb{Z}$. In this case, $E(T) = M(T) = \emptyset$. The first question asks for a $\emptyset = E(T) \subsetneq M(T)$ example.

In the locally-compact metrizable case, the set of positive invariant measures $\mu$ with $0 \leq \mu(X) \leq 1$ is compact (weak* topology) with extremals with total measures equal to $0$ or $1$. That is, according to Krein-Milman Theorem, if $M(T) \neq \emptyset$, then $E(T) \neq \emptyset$. So, an answer to Question 1 is not supposed to be locally-compact metrizable.

Question

1. Examples of continuous transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).
   Notice that the measures considered are defined over the Borel sets of $X$.

2. Example of a dynamical system where the following inequality is strict: $$\sup_{m \in E(T)} h_m(T) < \sup_{\mu \in M(T)} h_\mu(T)$$.

Background

Consider $T(x) = x + 1$ over the set of integers $\mathbb{Z}$. In this case, $E(T) = M(T) = \emptyset$. The first question asks for a $\emptyset = E(T) \subsetneq M(T)$ example.

In the locally-compact metrizable case, the set of positive invariant measures $\mu$ with $0 \leq \mu(X) \leq 1$ is compact (weak* topology) with extremals with total measures equal to $0$ or $1$. That is, according to Krein-Milman Theorem, if $M(T) \neq \emptyset$, then $E(T) \neq \emptyset$. So, an answer to Question 1 is not supposed to be locally-compact metrizable.

[Edit: The question only makes sense if the $\sigma$-algebra is fixed. So the post was edited, making $X$ a topological space, $T$ continuous and the $\sigma$-algebra is the family of Borel sets.]

Question

1. Examples of measurable transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).

2. Example of a dynamical system where the following inequality is strict:

   $\sup_{m \in E(T)} h_m(T) < \sup_{\mu \in M(T)} h_\mu(T)$.

Background

Consider $T(x) = x + 1$ over the set of integers $\mathbb{Z}$. In this case, $E(T) = M(T) = \emptyset$. The third question asks for a $\emptyset = E(T) \subsetneq M(T)$ example.

In the locally-compact metrizable case, the set of positive invariant measures $\mu$ with $0 \leq \mu(X) \leq 1$ is compact with extremals with total measures equal to $0$ or $1$. That is, according to Krein-Milman Theorem, if $M(T) \neq \emptyset$, then $E(T) \neq \emptyset$. So, an answer to Question 1 is not supposed to be locally-compact metrizable.

Question

1. Examples of measurable transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).

2. Example of a dynamical system where the following inequality is strict:

   $\sup_{m \in E(T)} h_m(T) < \sup_{\mu \in M(T)} h_\mu(T)$.

Background

Consider $T(x) = x + 1$ over the set of integers $\mathbb{Z}$. In this case, $E(T) = M(T) = \emptyset$. The first question asks for a $\emptyset = E(T) \subsetneq M(T)$ example.

In the locally-compact metrizable case, the set of positive invariant measures $\mu$ with $0 \leq \mu(X) \leq 1$ is compact (weak* topology) with extremals with total measures equal to $0$ or $1$. That is, according to Krein-Milman Theorem, if $M(T) \neq \emptyset$, then $E(T) \neq \emptyset$. So, an answer to Question 1 is not supposed to be locally-compact metrizable.