# A theory of bifurcation of braids ?

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.

• Toby Hall of Liverpool (also Phil Boyland?) has written fairly extensively on this topic in the early 1990s, I think. – Anthony Quas Jan 15 '13 at 22:43
• This is more of a "words you use resemble words I use" sort of response, but you never know. Look at: The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V, J. Group Theory, 10 (2007), 757-788, and The Algebra of Strand Splitting. II. A Presentation for the Braid Group on One Strand, Int'l. J. of Algebra and Computation, 16 (2006), 203-219. – Matt Brin Jan 15 '13 at 23:22
• Stupid question maybe, but how does a "punctured disk" qualify as a compact surface? – Loïc Teyssier Jan 16 '13 at 2:30
• Anthony: I am aware of the work of two gentlemen you mentioned, and have in fact used plenty of their results in my work. But they haven't touched the issue of braid bifurcation as far as I can tell. Matt: Thanks for the links you provided. Indeed, it seems we are speaking somewhat similar language but I would have to take a closer look. Loic: Good catch. I was thinking of disks with finite sized holes with boundary (i.e. for n=1 it would be a closed annulus) – Piyush Grover Jan 16 '13 at 15:34

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 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 about the analytical classification, continued in this paper 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 and also that one, 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 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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• Thanks very much Loic. These look like some very interesting and relevant papers. Appreciate your help! Time to brush up my complex analysis and dive in. – Piyush Grover Jan 16 '13 at 15:38