Let $\phi$ be a diffeomorphism of the unit disk $D^2$, fixed on the boundary, and suppose that $Q$ is a finite subset of the interior permuted by $\phi$. The isotopy class of $\phi$ relative to $Q$ and relative to the boundary determines a conjugacy class in a braid group $B_n$ where $n$ is the cardinality of $Q$.

One says that a conjugacy class of braid $b\in B_n$ forces another conjugacy class $b' \in B_m$ (where possibly $n$ is not equal to $m$) if every diffeomorphism $\phi$ representing $b$ permutes some finite subset $P$ in the interior of $D^2$ in such a way that the isotopy class of $\phi$ relative to $P$ represents $b'$.

A function from conjugacy classes in (all) braid groups to the non-negative reals is monotone if it can only go down under braid forcing; i.e. if the value of the function on $b'$ is less than or equal to its value on $b$ as above.

One way to define such a monotone invariant is to define some dynamical invariant of the conjugacy class of a diffeomorphism, and to take the infimum over all representatives. Since monotone braid classes give rise to inclusion of representative diffeomorphisms, such functions are necessarily monotone. One well-known (nontrivial) example is the (topological) entropy. Are there any other dynamically defined monotone invariants? What if one "stiffens" the structure, eg. by restricting to area-preserving diffeomorphisms?