Question

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.]

Answer 1

To answer question 1, I think there will always be a measure for which $T$ is ergodic.

Rotation of the unit circle $T(z) = az$ is measure preserving, (with Haar measure) and ergodic only when $a$ is not a root of unity. (Walters, P. Introduction to Ergodic Theory, Theorem 1.8).

However, we can still define a measure for which $T$ is ergodic. Say $\mu(\{a^k\}) = 1/n$ where $a$ an $n$-th root of unity, $k=0,\ldots, n-1$. Then the $T$-invariant sets either have measure 0 or 1.

When there are periodic points we can use the above argument to create an ergodic measure.

The argument is not very different in the absence of periodic points. For any measure preserving system $(X,\mathcal B, \nu)$ take any element $a \in X$ and define $orb_T(a) = \{T^na: n \in \mathbb Z\}$. Let $\mu$ be a probability measure on $\mathcal B / orb_T(a)$ with $\mu(0 + orb_T(a)) = 1$ and zero otherwise. Then $T$ is ergodic on $(X, \mathcal B/ orb_T(a), \mu)$.