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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?

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Naive question: I draw a picture of a braid on n strands and then erase n-m of the strands. Is the resulting m-strand braid "forced: by the original braid in your sense? –  JSE Nov 12 '09 at 5:50
    
Yes, if the strands you erase (or, equivalently, keep) are a union of orbits of the diffeomorphism. In other words, if $Q$ is a finite subset permuted by $\phi$, and $P$ is a subset of $Q$ invariant under $\phi$ then erasing (or keeping) $Q$ gives you a new braid (which is forced by the original braid). –  Danny Calegari Nov 12 '09 at 6:00
    
(So for instance, the "empty braid" is forced by everything) –  Danny Calegari Nov 12 '09 at 14:56
    
You might have a look at Ghys' 2006 ICM talk: icm2006.org/proceedings/Vol_I/15.pdf –  Ian Agol Oct 1 '11 at 2:05
    
@Agol: essentially all the invariants discussed in Ghys' paper (eg helicity, Ruelle invariant, Calabi quasimorphism etc.) as well as many variations (Polterovich, Py, etc.) are obtained by taking some local topological invariant of the dynamics on finitely many points, and integrating it over the degrees of freedom of the choice of the finitely many points wrt an invariant measure. If you take any braid type and "shrink" the dynamics down to be concentrated in a very small disk, the value of these invariants typically goes to zero. Did you have any specific part of the paper in mind? –  Danny Calegari Oct 1 '11 at 6:33
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