Let be a homeomorphism on a compact metric space . Then is said to be -expansive, if there exists such that for any two points , if for all , then . The constant is called the expansive constant of .

Similarly one can define -expansiveness if is not invertible. An interesting phenomenon observed by Schwartzman states that

**Theorem.** A homeomorphism cannot be -expansive (unless is finite).

This result was reported in Gottschalk–Hedlund’s book *Topological Dynamics* (1955), and a proof was given in King’s paper *A map with topological minimal self-joinings in the sense of del Junco* (1990). Below we copied the proof from King’s paper.

**Proof.** Suppose on the contrary that there is a homeo on that is -expansive. Let be the -expansive constant of , and .

It follows from the -expansiveness that is a finite number. Pick such that whenever .

*Claim.* If , then for any .

Proof of Claim. If not, we can prolong the -string since .

Recall that a pair is said to be -proximal, if for some . The upshot for the above claim is that any -proximal pair is -indistinguishable: for all .

Cover by open sets of radius , and pick a finite subcover, say . Let be a subset consisting of distinct points. Then for each , there are two points in share the room , say , and . Pick a subsequence such that and . Clearly , and . Hence the pair is -proximal and -indistinguishable. This contradicts the -expansiveness assumption on . QED.

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