Let $(X,mathcal{A},m)$ be some probability space where $m=frac{1}{p}sum_{j=0}^{p-1}delta_{f^jx}$ for some fixed $xin X$ that is $p$-periodic with respect to the measure-preserving transformation $fcolon Xto X$, i.e., $f^p(x)=x$. Here the $delta_{T^jx}$ is the Dirac-measure.

Now, let $A={A_1,ldots,A_k}$ be some finite measurable partition of $X$ and define

$$

bigvee_{i=0}^{n}f^{-i}A:=left{bigcap_{i=0}^nf^{-i}A_{j_i}: A_{j_i}in Aright}.

$$

I would like to verify that for the entropy $h(f,A)$ of $f$ with respect to $A$ one has $h(f,A)=0$.

To this end, one needs to argue that

$$

h(f,A)=lim_{ntoinfty}frac{1}{n}Hleft(bigvee_{i=0}^{n}f^{-i}Aright)=0,

$$

where $Hleft(bigvee_{i=0}^{n}f^{-i}Aright)$ is the entropy of the partition $alpha_n:=bigvee_{i=0}^{n}f^{-i}A$ which is defined as

$$

H(alpha_n)=-sum_{alphainalpha_n}m(alpha)log m(alpha).

$$

Here’s what I’ve tried so far:

Without loss of generality, we can assume that $n>p$. Due to the periodicity of $x$, we have

$$

xin A_{j_0},quad fxin A_{j_1},quadldots,quad f^{p-1}xin A_{j_{p-1}},quad f^pxin A_{j_0},quad,f^{p+1}xin A_{j_1},quadldots

$$

for some unique $A_{j_i}in A, i=0,1,2,ldots$.

Thus, for $0leq ileq p-1$,

$$

delta_{f^ix}(alpha)=begin{cases}1, & alpha=A_{j_i}cap f^{-1}A_{j_{i+1}}capldotscap f^{-(p-1)}A_{j_{i+p-1}}cap f^{-p}A_{j_i}cap f^{-(p+1)}A_{j_{i+1}}capldotscap f^{-n}A_{j_{i+n}}\0, & textrm{otherwise}end{cases}

$$