Cases where
$sup_{\mu \in E(T)} h_\mu(T)
\neq
\sup_{\mu \in M(T)} h_\mu(T)$.
Background
For a topological space $X$,
let $T: X \to X$ be a continuous application.
Then, call the set of $T$-invariant probability measures
$M(T)$, and call the set of $T$-ergodic (probability) measures
$E(T)$.
It is evident that $E(T) \subset M(T)$.
But it may happen that $M(T) = \emptyset$.
For example, take $X = \mathbb{R}$ and $T(x) = x+1$.
Since an ergodic measure is invariant, it is immediate that
$
\begin{equation*}
\sup_{\mu \in E(T)} h_\mu(T)
\leq
\sup_{\mu \in M(T)} h_\mu(T).
\end{equation*}
$
The question is whether equality holds or not.
When $X$ is compact, it is well known that equality holds.
In this case, it is a consequence of Jacobs' Theorem, which
states that for any $\mu \in M(T)$, there exists a measure
$\tau$, over the set $E(T)$, such that
$
\begin{equation*}
h_\mu(T)
=
\int_{E(T)} h_m(T) d\tau(m).
\end{equation*}
$
When $X$ is compact (locally compact, in fact), the above equation is a consequence of Choquet Representation Theorem and the Krein-Milman Theorem. (See, for example, Theorem 8.4 from Walters, P. An Introduction to Ergodic Theory)
Now, when $X$ is not necessarily compact, but it is a Borel subset of a compact metrizable set $\widetilde{X}$, Pesin and Pitskel' argue in their Topological Pressure and the Variational Principle for Noncompact Sets, at the end of page 310: (I will rename the spaces and applications in order to conform to this post's notation.)
We may assume that measure $\mu$ is ergodic. In fact, consider the partition $\eta$ of $X$ into ergodic components
$X_s,\, s \in S$, of measure $\mu$. Denote by$\mu_s$the measures on$X_s$(then$T * \mu_s = \mu_s$), and by $\nu$ the measure on the quotient space $X / \eta$. Then$h_\mu(T) = \int_{Y/\eta} h_m(T) d\nu(m)$.
As far as I understand, $Y/\eta$ is just the same as $E(T)$, since each ergodic component is associated with an ergodic measure. And for the same reason, $\nu$ is just our $\tau$. So, what is being stated is the validity of
$
\begin{equation*}
h_\mu(T)
=
\int_{E(T)} h_m(T) d\tau(m),
\end{equation*}
$
which in turns implies the equality
$
\begin{equation*}
\sup_{\mu \in M(T)} h_\mu(T)
=
\sup_{\mu \in E(T)} h_\mu(T).
\end{equation*}
$
In Pitskel' and Pesin's paper, $T$ is not even supposed to be the restriction
to $X$ of a continuous transformation
$\widetilde{T}: \widetilde{X} \to \widetilde{X}$.
Questions
How do I prove that when $X$ is a Borel subset of a compact metrizable space
$\widetilde{X}$and $T$ is a continuous application$T: X \to X$, then for any$\mu \in M(T)$, there exists a measure $\tau$ over $E(T)$ such that$h_\mu(T) = \int_{E(T)} h_m(T) d\tau(m)$?In case the answer to question "1" is negative, is there a prove for the specific case where $T$ is the restriction of a continuous application
$\widetilde{T}: \widetilde{X} \to \widetilde{X}$?Do you know nice examples of transformations of measurable spaces where
$E(T) = \emptyset$while$M(T) \neq \emptyset$?
PS: This is my first post to MathOverflow. This is really exciting! :-)
