Prove that the topological entropy $T$ is zero.

Let $T$ be the isometry of a metric compact with a dense orbit. Prove that the topological entropy $T$ is zero.

What happened now: $T$ is the isometry of a metric compact, that is, for any two points of a compact with metric d, we have $d(T(a), T(b)) = d(a, b)$. Since by the condition with a dense orbit, then $T$ is a topologically transitive map. We have that $d(T^ i (a), T^i (b)) = . . . = d(T ^2 (a), T^2 (b)) = d(T(a), T(b)) = d(a, b) $. Help to prove. Happy New Year)

Hint: There are multiple ways of calculating/defining topological entropy, in particular there are ways of calculating it via the so-called Bowen metrics (I mentioned these metrics in a previous answer here: https://math.stackexchange.com/a/4326733/169085). Say we want to use the spanning numbers w/r/t Bowen metrics. We'll need to take the logarithm of the $(n,\epsilon)$-spanning number; divide it by $n$ and then take limits. But observe that since the transformation is an isometry these spanning numbers are independent of $n$, as you've observed.

(Also, this statement is true without the point transitivity assumption.)