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No. Let $X$ be a closed interval and let $\phi:X\to X $ be an order-preserving bijection that fixes only the endpoints.
Edit: This is an answer to a trivial question which is not the intended one.
No. Let $X$ be a closed interval and let $\phi:X\to X $ be an order-preserving bijection.
No. Let $X$ be a closed interval and let $\phi:X\to X $ be preserve thean order and fix only the two endpoints-preserving bijection.
No. Let $X$ be a closed interval and let $\phi:X\to X $ be preserve the order and fix only the two endpoints.
