First I have to give the background to my question:
Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the map $T\colon X\to X$ which describes the following dynamics: For $x\in X$, which is a bi-infite sequence, let $x(i)$ denote the i-th position. If $x(i)=1$ then in the next time step, $x(i)=2$. Similarly, a 2 becomes a 0 and $x(i)=0$ becomes $x(i)=1$ if at least one of the neighbour positions $x(i-1)$ and $x(i+1)$ is 1.
Now consider the following set $Y$, consisting of all $x\in X$ with the following property: There exists a $n\in\mathbb{Z}\cup\left\{-\infty,\infty\right\}$ such that
at position $n$ and to the left of $n$,
every 1 has a 2 to its left
every 2 has a 0 to its left
and every 0 has a 0 or a 1 to its left
and to the right of $n$,
every 1 has a 2 to its right
every 2 has a 0 to its right
and every 0 has a 0 or a 1 to its right.
Moreover, consider the set $Y'$ consisting of all $x\in\left\{0,r,l\right\}^{\mathbb{Z}}$ with the following property:
There exists a $n\in\mathbb{Z}\cup\left\{-\infty,\infty\right\}$ such that
at position $n$ and to the left of n
- there are only r's and 0s with at least two 0s between r's
and to the right of $n$,
- there are only l's and 0's with at least two 0s between l's.
Let $T'$ map $\left\{0,r,l\right\}^{\mathbb{Z}}$ to itself by having the r's move right, the l's move left, and an r and an l annihilate each other when they meet or cross.
There is a natural map $U$ from $Y\to Y'$ as follows. Let $$ U(\eta)(x)=\begin{cases}r, & \text{if }\eta(x)=2\text{ and }\eta(x+1)=1\text{ or }2\\l, & \text{if }\eta(x)=2\text{ and }\eta(x-1)=1\text{ or }2\\0, & \text{otherwise}\end{cases} $$
--- Now to my question:
Because $U$ is a continuous surjection, for the topological entropy of $T$ and $T'$ it is $$ h(T)\geqslant h(T'). $$
(Here $h(T)$ resp. $h(T')$ are topological entropy of $T$ resp. $T'$ using the definition of topological netropy via open coverings.)
Now, concerning Theorem 17 on page 409, it is $$ h_d(T)\leq h_e(T')+\sup_{y'\in Y'}h_d(T,U^{-1}(y')), $$ where $h_d(T)$ and $h_e(T')$ are topological entropy defined by Bowen.
Is it true that $$ \sup_{y'\in Y'}h_d(T,U^{-1}(y'))=0? $$
(If yes, then $$ h(T)=h(T'), $$ because $Y$ and $Y'$ are compact metric spaces and $T$ and $T'$ are continuous and therefore $h(T)=h_d(T)$ and $h(T')=h_e(T')$.)
As a metric d on $Y$ I think we can use $$ d(x,y)=\begin{cases}2^{-k}\text{ with k maximal so that }x_{[-k,k]}=y_{[-k,k]}, & x\neq y\\0, & x=y\end{cases} $$
And as metric $e$ on $Y'$ the same.
Hope, you can help me, because I am rather helpless.