  For the Lagrange spectrum and other applications, we determine the smallest
accumulation point of binary sequences that are maximal in their shift orbits.
This problem is trivial for the lexicographic order, and its solution is the
fixed point of a substitution for the alternating lexicographic order. For
orders defined by cylinders, we show that the solutions are $S$-adic sequences,
where $S$ is a certain infinite set of substitutions that includes Sturmian
morphisms. We also consider a similar problem for symmetric ternary shifts,
which is applicable to the multiplicative version of the Markoff-Lagrange
spectrum.
