4

The unique measure of maximal entropy $\mu_f$ supported on the Julia set of a rational map $f$ of degree $d \geq 2$ is indeed the unique balanced measure for $f$, i.e., the only probability measure $\mu$ not charging the exceptional set and satisfying $f^*\mu =d \cdot \mu$. As you already noticed in the comments, uniqueness of a measure with this property ...

3

In general $F$ is not ergodic. A very simple example can be constructed as follows: let $X=\mathbb{Z}_3=\{0,1,2\}$ and $\mu =1/3(\delta_0+\delta_1+\delta_2)$ and $T(x):=x+1$. This is an ergodic system. Let us define $Y:=\{0,1\}$ and $R\equiv 3$. Since $F=T^3=id$, it is not ergodic.

2

One should always keep in mind that the natural substrate of ergodic theory is Lebesgue-Rokhlin (aka Lebesgue or standard) measure spaces which enjoy a lot of properties not necessarily present in general measure spaces. One of these properties is an explicit description of homomorphisms of such spaces obtained by Rokhlin in his 1949 paper, which, in ...

2

If $K$ is a real quadratic field of degree $n$ Truelsen (see https://arxiv.org/abs/0706.4239) showed QUE for for Eisenstein series on the arithmetic quotient $\text{PSL}_2(O_K)\backslash (H^2)^n$. I am not aware for a reference dealing with arbitrary number fields $K$ even if it should be possible to do this.
A true higher rank example has been worked out ...

2

Begin by thinking about the dual automorphism to a hyperbolic toral automorphism. You can project the dual group onto an "expansive" direction. Any relation of the sort you are looking at can be shifted by the automorphism so that all powers are negative except for one large positive one. In the projection, this would mean that a huge number is the sum of ...

2

The measure of maximal entropy is the unique measure that is "fully invariant" in your sense. I believe that this already follows from the original proofs - indeed, it is well-known that if you take a point mass at some non-exceptional point, and keep pulling back, you will converge to the measure of maximal entropy. This should be enough to deduce the claim....

1

The way that the paper by Freire-Lopes-Mañé makes sense of $f^*\mu=d\mu$ is the following: ''For any Borel subset $A$ of $\Bbb{C}_\infty$ with $f\restriction_A$ injective, one has $\mu(f(A))=d.\mu(A)$.'' (See p. 46 of this paper.)
One observation is that such an ergodic measure $\mu$ is either supported on the Julia set or is one of those measures with ...

1

Now I figure out that the answer of my question is false. In the paper ``Topological mixing and uniquely ergodic systems", Lehrer proved any measure-preserving system has a topologcally mixing strictly ergodic topological model, which implies that there exists a topologcally mixing strictly ergodic system $(X,T)$ such that with the unique invariant measure $\...

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