I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional time dependent ODE on the disk, forward in time. Let us assume that the ODE system (and hence the resulting diffeomorphism) is dependent on one bifurcation parameter.


Also assume that we select some 'distinguished' periodic orbits of this system, and start varying the bifurcation parameter. For each value of this parameter, we can form a braid, where 'world-line' of each periodic orbit (in 2+1 space) is a strand. I am interested in exploring the relationship between the different braids that are formed when we vary the parameter, and especially the behavior near a bifurcation point of any one (or more) periodic points that we selected. Are there any results that shed some light on this ? I have been advised to look for connections with winding number of the periodic orbits, but so far I haven't been able to find any relevant literature.


The motivation comes from the fact that braids formed above encode quite a bit of information, including but not limited to, the topological entropy of the flow. One can obtain lower bounds on topological entropy by invoking the Thurston-Nielsen theorem, and so on.


Thanks for any insight.



A theory of bifurcation of braids ?
Here is a couple of references regarding the holomorphic world, although they do not represent an answer to your question per se (but I'm afraid this is too long a comment, and might anyhow interest people interested in your question). 


First there is [a work](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=moussu&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=6&mx-pid=2135673) by Cano, Moussu and Sanz where they study the way real trajectories attached to some complex ODE are entangled.


Next, in the following links, the braid structure is not studied as such, but I think you may perform the study (in this special context) by using the tools and constructions developped below.


* In the context of the bifurcation of holomorphic diffeomorphisms of a disk, there is [this one](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=192734&vfpref=html&r=17&mx-pid=2316241) about the analytical classification, continued in [this paper](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=192734&vfpref=html&r=8&mx-pid=2593608) where the moduli space is completely identified.
* Since you are interested in the suspension itself (what you describe as the ODE), maybe you could have a look at [this one](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=192734&vfpref=html&r=11&mx-pid=2483224) and also [that one](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=192734&vfpref=html&r=19&mx-pid=2223354), which deal with the complex suspension of the diffeomorphisms appearing above, the first one describing more particularly the underlying geometry. You retrieve a real suspension by lifting in the complex solutions a circle included in one fo the separatrices: the subsequent holonomy is precisely the diffeomorphism you started from.
* I also point out [this reference](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=681008&vfpref=html&r=3&mx-pid=2647978) for (complex) two-dimensional holomorphic dynamics, which really is also a deformation of one-dimensional diffeomorphisms.


Hope this somehow helps you in tackling your problem.


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