Dtyjlui

Am trying to prove the following :

Let ($X,T$) a uniquely ergodic topological system.We suppose that the uniquely $T-$invariant measure $mu$ satisfies the property :$mu(U)>0$ for every non-empty open set $Usubseteq X$.Then show under these conditions that the pair $(X,T)$ is minimal.

Let me clarify some definitions:

1)Topological System is the pair $(X,T)$ where $X $ is a compact metric space and $T:Xto X$ is a continuous transformation.

2)Minimal topological System is the pair $(X,T)$ where there does not exist a closed set $emptyset neq F subseteq X$ such that $Fneq X$ and $T(F)subseteq F$.

I tried to approach this by showing that every $x in X$ has dense orbit but i only showed that
$$mubiggl(bigl{x in X: overline{{T^n(x):n in mathbb{N}_0}}=Xbigr}biggr) =1$$

and i dont know has to use the assumption that $mu$ is uniquely determined by being $T-$invariant.

Do you have any ideas? Thanks in advance !

Ok guys i think proved it, i used the following preposition which is equivalent for a dynamical topological pair $(X,T)$ to be uniquely ergodic :

For every continuous function $fin C(X)$
$$frac{1}{n}sum_{k=0}^{n-1}f(T^k(x)) to int fdmu tag 1$$
for every $xin X$ where $mu$ is the uniquely determined $T-$invariant Borel probability measure.

Now i can prove the the desired result as follows:

Suppose we have a set $F$ which closed, $T-$invariant $bigl(T(F)subseteq Fbigr)$ and $F neq X$ then $Xsetminus F$ is non empty and open. Let $x_1 in Xsetminus F$, then there is an $epsilon>0$ such that
$$E=hat{B}(x_1,epsilon) = {xin X: d(x,x_1)leq epsilon} subseteq Xsetminus F$$
From our assumptions we get that $mu(E)>0 $ since $hat{B}(x_1,epsilon)$ contains the open ball $B(x_1,epsilon).$

Now we are trying to find the crucial continuous function $f$ in order to use $(1)$.Since we have two closed and disjoint sets $F,E$ we define
$$f(x) = frac{dist(x,F)}{dist(x,F)+dist(x,E)}$$
then we know that

$alpha)$ $f$ is continuous

$beta)$ $0leq f leq 1$

$gamma$) $f|F =0$ and $f|E=1$

So if we combine these facts and $(1)$ we get that
$$frac{1}{n}sum_{k=0}^{n-1}fbigl(T^k(x)bigr) to int fdmu=int_{Xsetminus F}f dmu geq int_{E}fdmu geq mu(E)>0 tag 2$$
for every $xin X$.

But now (2) tells us that $F$ must be the empty set since if we can find $x^*in F$ since $T(F) subseteq F$ we get that $T^k(x^*) in F$ for every k. In other words
$$frac{1}{n}sum_{k=0}^{n-1}fbigl(T^k(x^*)bigr) =0 $$
for every $n in mathbb{N}$ which contradicts $(2)$ since it holds for every $xin X$.

So, for every $F$ closed which is $T-$invariant $bigl(T(F)subseteq Fbigr)$ it must be either $F= emptyset$ either $F=X$.So, $(X,T)$ must be minimal.

Is this correct? thanks again !