# Values appearing with density in an ergodic system

Values appearing with density in an ergodic system

Let $$(X,\mu)$$ be a probability space with invertible, measure preserving, totally-ergodic map $$T:X \to X$$. ($$(X,\mu,T)$$ is a $$\mathbb{Z}$$ dynamical system). Suppose $$f:X \to \mathbb{R}$$ is a measurable function such that $$\mu(x \colon f(x) = a) = 0$$ for all $$a \in \mathbb{R}$$.

Let

$$D(x) = \left\{ a \in \mathbb{R} \colon \liminf_{n \to \infty} \frac{ \# \{0 \leq k < n \colon f(T^{k}x) = a \}}{n} > 0 \right\}$$

be the set of numbers that appear with density. Is $$D(x) = \emptyset$$ with probability $$1$$? Are there standard conditions on $$(X,\mu,T)$$ like mixing, positive entropy and so on that guarantee that $$D(x) = \emptyset$$ with probability $$1$$? One such is the example where $$X = [0,1]^\mathbb{Z}$$ $$\mu = \mu_0^{\otimes \mathbb{Z}}$$ where $$\mu_0$$ is the uniform measure on $$[0,1]$$, and $$T$$ is the shift map.

Speculation: Obviously this is true for all the rationals; i.e., $$D(x) \cap \mathbb{Q} = \emptyset$$ with probability 1. So maybe I need some sort of continuity for this to have a chance to be true? I'm in the fairly general setting of a shift system, and do have some flexibility in the way I formulate my problem. My only constraint is that $$f$$ should have the right "finite dimensional distributions" that are specified, and $$X$$ is a factor of a Bernoulli system ($$\mu$$ is a product measure). So perhaps I can construct $$X$$ and $$f$$, and put a topology on it such that it is continuous?

• I suppose you wanted $\#\{0\le k<n\colon f(T^{k}x)=a\}$ instead of $\{0\le k<n\colon f(T^{k}x)=a\}$. – Iosif Pinelis Feb 14 at 18:50
• @IosifPinelis yup, I fixed, thanks! – arjun Feb 14 at 19:28

Indeed $$D(x)$$ is empty for almost all $$x$$. It suffices to show that for all integers $$b,\ell>0$$, the set $$D_b(x,\ell) = \left\{ a \in [b,b+1) \colon \liminf_{n \to \infty} \frac{ \# \{0 \leq k < n \colon f(T^{k}x) = a \}}{n} > \frac{1}{\ell} \right\}$$ is empty for almost all $$x$$, and then take a countable union over $$b,\ell$$. Fix $$m$$ large enough so that $$\max_{1 \le j \le m} \mu(f^{-1}[b+(j-1)/m,b+j/m)) <1/\ell \,.$$ (Such $$m$$ exists by the assumption that $$\mu(x \colon f(x) = a) = 0$$ for all $$a \in \mathbb{R}$$.)
write $$E_b (\ell,j) = \left\{ x \in X \colon \liminf_{n \to \infty} \frac{ \# \{0 \leq k < n \colon f(T^{k}x)\in [b+(j-1)/m,b+j/m) \}}{n} > \frac{1}{\ell} \right\} \,.$$ the Birkhoff Ergodic Theorem implies that $$\mu[ E_b (\ell,j)]=0$$ for all $$b,\ell,j$$. Since $$\{x :\, D_b(x,\ell)\ne \emptyset\} \subset \bigcup_{j=1}^m E_b(\ell,j) \,,$$ we conclude that $$\mu\{x :\, D_b(x,\ell)\ne \emptyset\} =0$$.