Conjugacy of real diffeomorphisms. A survey

O’Farrell, A.; Roginskaya, M.

doi:10.1090/S1061-0022-2010-01130-0

Conjugacy of real diffeomorphisms. A survey
HTML articles powered by AMS MathViewer

by A. G. O’Farrell and M. Roginskaya

St. Petersburg Math. J. 22 (2011), 1-40

DOI: https://doi.org/10.1090/S1061-0022-2010-01130-0

Published electronically: November 16, 2010

PDF | Request permission

Abstract:

Given a group $G$, the conjugacy problem in $G$ is the problem of giving an effective procedure for determining whether or not two given elements $f,g\in G$ are conjugate, i. e., whether there exists $h\in G$ with $fh=hg$. This paper is about the conjugacy problem in the group $\operatorname {Diffeo}(I)$ of all diffeomorphisms of an interval $I\subset \mathbb {R}$.

There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.

References

V. Afraimovich, W. S. Liu, and T. Young, Conventional multipliers for homoclinic orbits, Nonlinearity 9 (1996), no. 1, 115–136. MR 1374001, DOI 10.1088/0951-7715/9/1/004
Patrick Ahern and Jean-Pierre Rosay, Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables, Trans. Amer. Math. Soc. 347 (1995), no. 2, 543–572. MR 1276933, DOI 10.1090/S0002-9947-1995-1276933-6
G. R. Belitskiĭ, Smooth classification of one-dimensional diffeomorphisms with hyperbolic fixed points, Sibirsk. Mat. Zh. 27 (1986), no. 6, 21–24 (Russian). MR 883578
William E. Boyce and Richard C. DiPrima, Elementary differential equations and boundary value problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0179403
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
Jonathan Lubin, Non-Archimedean dynamical systems, Compositio Math. 94 (1994), no. 3, 321–346. MR 1310863
Arnold B. Calica, Reversible homeomorphisms of the real line, Pacific J. Math. 39 (1971), 79–87. MR 307144
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math. 22 (1970), 165–173. MR 267589
Nancy Kopell, Commuting diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 165–184. MR 0270396
Edward Kasner, Conformal classification of analytic arcs or elements: Poincaré’s local problem of conformal geometry, Trans. Amer. Math. Soc. 16 (1915), no. 3, 333–349. MR 1501016, DOI 10.1090/S0002-9947-1915-1501016-6
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Marek Kuczma, Bogdan Choczewski, and Roman Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990. MR 1067720, DOI 10.1017/CBO9781139086639
John N. Mather, Commutators of diffeomorphisms, Comment. Math. Helv. 49 (1974), 512–528. MR 356129, DOI 10.1007/BF02566746
—, Commutators of $C^r$ diffeomorphisms of the real line (unpublished preprint, privately communicated).
John N. Mather, On Haefliger’s classifying space. I, Bull. Amer. Math. Soc. 77 (1971), 1111–1115. MR 283817, DOI 10.1090/S0002-9904-1971-12886-0
John N. Mather, Integrability in codimension $1$, Comment. Math. Helv. 48 (1973), 195–233. MR 356085, DOI 10.1007/BF02566122
H. Mirkil, Differentiable functions, formal power series, and moments, Proc. Amer. Math. Soc. 7 (1956), 650–652. MR 79064, DOI 10.1090/S0002-9939-1956-0079064-7
Andrés Navas, Groupes résolubles de difféomorphismes de l’intervalle, du cercle et de la droite, Bull. Braz. Math. Soc. (N.S.) 35 (2004), no. 1, 13–50 (French, with English and French summaries). MR 2057043, DOI 10.1007/s00574-004-0002-2
Andrés Navas, Grupos de difeomorfismos del círculo, Ensaios Matemáticos [Mathematical Surveys], vol. 13, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007 (Spanish, with English summary). MR 2394157
Anthony G. O’Farrell, Conjugacy, involutions, and reversibility for real homeomorphisms, Irish Math. Soc. Bull. 54 (2004), 41–52. MR 2138430
Anthony G. O’Farrell, Composition of involutive power series, and reversible series, Comput. Methods Funct. Theory 8 (2008), no. 1-2, 173–193. MR 2419471, DOI 10.1007/BF03321681
A. G. O’Farrell and M. Roginskaya, Reducing conjugacy in the full diffeomorphism group of $\mathbb R$ to conjugacy in the subgroup of orientation-preserving maps, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 360 (2008), 231–237; English transl., J. Math. Sci. 158 (2009), 895–898.
Anthony G. O’Farrell and Ian Short, Reversibility in the diffeomorphism group of the real line, Publ. Mat. 53 (2009), no. 2, 401–415. MR 2543857, DOI 10.5565/PUBLMAT_{5}3209_{0}5
Joel W. Robbin, Unfoldings of discrete dynamical systems, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 421–486. MR 776878, DOI 10.1017/S0143385700002558
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
F. Sergeraert, Feuilletages et difféomorphismes infiniment tangents à l’identité, Invent. Math. 39 (1977), no. 3, 253–275 (French). MR 474327, DOI 10.1007/BF01402976
Shlomo Sternberg, Local $C^{n}$ transformations of the real line, Duke Math. J. 24 (1957), 97–102. MR 102581
G. Szekeres, Regular iteration of real and complex functions, Acta Math. 100 (1958), 203–258. MR 107016, DOI 10.1007/BF02559539
Floris Takens, Normal forms for certain singularities of vectorfields, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 2, 163–195 (English, with French summary). MR 365620
S. M. Voronin, Analytic classification of germs of conformal mappings $(\textbf {C},\,0)\rightarrow (\textbf {C},\,0)$, Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 1–17, 96 (Russian). MR 609790
Hassler Whitney, Differentiable functions defined in closed sets. I, Trans. Amer. Math. Soc. 36 (1934), no. 2, 369–387. MR 1501749, DOI 10.1090/S0002-9947-1934-1501749-3
Jean-Christophe Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle, Astérisque 231 (1995), 89–242 (French). Petits diviseurs en dimension $1$. MR 1367354
Todd R. Young, $C^k$ conjugacy of $1$-D diffeomorphisms with periodic points, Proc. Amer. Math. Soc. 125 (1997), no. 7, 1987–1995. MR 1372046, DOI 10.1090/S0002-9939-97-03783-0
H. Eynard, On the centralizer of diffeomorphisms of the half-line (to appear); arXiv:0811.1173.