The second main result of the paper is the following: Suppose that $X$ is a compact subset of $I$ which contains the endpoints of $I$. Suppose that $f$ and $g$ are continuous maps of $I$ to itself with $f(X)\subset X$, $g(X)\subset X$, and $f|_X=g|_X$. Suppose that for each component $K$ of $I\sbs X$, the restriction of $f$ to $K$ is monotonic. Then $h(f)\leq h(g)$.
Stripped down, that's "Suppose this. And suppose this, with this and this and this. And suppose this. Then it's safe to say this, anyway."
Surely my frequent exposure to all this positing and setting up of propositions, and the attendant necessity for certain sorts of conditions and positing before anything is said, or concluded, or arrived at, not to mention bevies of multiply-embedded if-then statements, can only reinforce the knee-jerk cogitatin', figuratin' 'Ff'lo. And perhaps it has something to do with why I want more and more to jump out of an airplane. Or stand in the pouring rain. Or why I seem to be really getting into exhausting myself physically.
Or perhaps not! I'm going to ---gasp--- stop thinking about it now.