It is well known that a continuous injection from $[0,1]$ to itself or a homeomorphism from a unit circle to itself has zero topological entropy. But what about continuous injections from unit circle to itself? Does it still have zero topological entropy? Note that injectivity is necessary otherwise we have the example of doubling map which has $\log 2$ entropy.
I suspect the proof of continuous injection from $[0,1]$ to itself or a homeomorphism from a circle to itself have zero topological entropy might still work with minor adjustment but I am not sure.
