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Conjugacy of real diffeomorphisms. A survey. (English) Zbl 1221.37084

St. Petersbg. Math. J. 22, No. 1, 1-40 (2011) and Algebra Anal. 22, No. 1, 3-56 (2010).
This paper provides a systematic review of the conjugacy problem for the group \(\text{Diffeo}(I)\) of all diffeomorphisms of an interval \(I\subset\mathbb{R}\). Explicitly, the question is whether for two elements of \(f,g\in \text{Diffeo}(I)\) there is an \(h\in \text{Diffeo}(I)\) so that the compositions \(fh\) and \(hg\) are equal. There has been a considerable amount of classical work on this subject, dealing particularly with special classes of mappings. The authors note that many results, arguments and techniques are known to experts in the field, but are difficult to find (or absent) in the literature. The current paper is an attempt to fill in the gaps. Some new results are also included.
A more recent result due to the authors is that the conjugacy problem in \(\text{Diffeo}(I)\) can be reduced to the corresponding problem in the (normal) subgroup \(\text{Diffeo}^+(I)\) of orientation-preserving diffeomorphisms of \(I\). This result and its proof are discussed briefly.
The conjugacy problem for \(\text{Diffeo}^+(I)\) is not a classical conjugacy problem of combinatorial group theory. \(\text{Diffeo}^+(I)\) does not have a countable presentation. Further, both \(\text{Diffeo}^+(I)\) and its family of conjugacy classes have the cardinality of the continuum. While an effective procedure to determine conjugacy for two maps \(f,g\in\text{Diffeo}^+(I)\) would be desirable, this is too much to hope for. (It turns out that the topological conjugacy map is already intractable.) So the goal is to provide a collection of classifying invariants that offer conceptual simplification of the conjugacy problem.

MSC:

37E05 Dynamical systems involving maps of the interval
20E45 Conjugacy classes for groups
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