Natural density and Poincare's recurrence

Question

Let $(X,\mathcal{B},\mu,T)$ be a dynamical system and let $A \in \mathcal{B}$ such that $\mu(A)>0$ and $\forall x \in X$ we define $$L_x=\{n \in \Bbb{N}|T^nx \in A\}$$

Prove that $\mu(\{x:\bar{d}(L_x)>0\})>0$ where $$\bar{d}(L_x)=\limsup_n \frac{|L_x \cap \{1,2...n\}|}{n}$$.

My first thought is that i have to use somewhere Poincare's recurrence theorem.

But i cannot understand the behavior of the density function.

From Poincare we just know that for almost every $x \in A$ we have that $L_x$ is infinite.

Can some give me a hint to solve this?

I clearly do not want a full solution.

I want hint to solve this only with Poincare's Theorem and the properties of the natural density function.(Not Ergodic Theorems)

Any help is appreciated.

Thank you in advance

Answer 1

For almost every $x$, the limit $$\lim_{N \to \infty} \frac{1}{N}\sum_{n=1}^{N} \chi_A(T^nx) = \lim_{N \to \infty} \frac{|L_x \cap [1,N]|}{N}$$ exists. Let $f^*(x)$ be the limit. It is a fact (see Birkhoff ergodic theorem - note this does apply to non-ergodic measure preserving systems) that $\int f^*(x)d\mu = \int \chi_A d\mu$. However, we only know this applies if $\mu(X)$ is finite (which I am assuming it is). Finish from here.

Answer 2

I am also assuming $\mu$ is a probability measure. For a measurable set $A$ with positive $\mu$-measure, define $A_0$ to be the set of points $x\in A$ such that $\overline{d}(L_x)=0$. Let $\chi_{A_0}(.)$ be the characteristic function over the set $A_0$ and for any $x\in X$, consider $$ F^+(x) = \limsup_{n\to +\infty} \frac{1}{n} \sum_{j=0}^{n-1} \chi_{A_0}\circ T^j(x). $$

From the definition of $A_0$, it is easy to prove that $F^+(x) = 0$ for every $x\in X$. Also observe that $$ F^-(x)= \liminf_{n\to + \infty} \frac{1}{n} \sum_{j=0}^{n-1} \chi_{A_0} \circ T^j(x) \geq 0. $$

Thus, for every $x\in X$, it is well defined

$$ F(x) = \lim_{n\to + \infty} \frac{1}{n} \sum_{j=0}^{n-1} \chi_{A_0} \circ T^j(x) = 0. $$

Of course for every $n\in\mathbb{N}$ and every $x\in X$, the function $ \frac{1}{n} \sum_{j=0}^{n-1} \chi_{A_0}\circ T^j(x)$ takes value between $0$ and $1$. By the dominated convergence theorem, one concludes that

$$ \mu(A_0) = \lim_{n\to+\infty} \int \frac{1}{n} \sum_{j=0}^{n-1} \chi_{A_0}\circ T^jd\mu= \int \lim_{n\to+\infty} \frac{1}{n} \sum_{j=0}^{n-1} \chi_{A_0}\circ T^jd\mu= \int F d\mu=0.$$

One concludes that $\mu$-almost every point in $A$ has a positive density.

Remark: The same result is true if you change $\limsup$ for $\liminf$ in the definition of density, but I don't know how to prove that without using some ergodic theorem.