Let $X$ be a compact and connected LOTS (linearly-ordered topological space). The topology on $X$ is the order topology.
Let $a<b\in X$.
I am trying to prove the following:
For every interval $(a,b)\subseteq X$ there exists a non-trivial order-preserving homeomorphism $f$ such that $f(x)=x$ for all $x\in X\smallsetminus (a,b)$.
I'm having trouble proving it. So I started thinking, maybe there is a counter example to this claim?
Any hints, ideas or references would be highly appreciated!
It is at least consistent that there be a counterexample. Specifically, assume the combinatorial principle $\Diamond$, which is consistent with $\mathsf{ZFC}$; then there is a rigid Suslin continuum $X$. $X$ is densely ordered and Dedekind complete, so it’s connected, and rigidity here means that it admits no non-trivial order-automorphism. A relatively accessible reference is Section $6$ of S. Todočević, ‘Trees and Linearly Ordered Sets’, in the Handbook of Set-Theoretic Topology, K. Kunen & J.E. Vaughan, eds., North Holland, $1984$.