## Reversibility in the diffeomorphism group of the real line.(English)Zbl 1178.37022

The article deals with reversible and strongly reversible elements of two groups: the group $${\text{Diffeo}} \, ({\mathbb R})$$ of infinitely differentiable homeomorphisms (diffeomorphisms) of $${\mathbb R}$$ and its subgroup $${\text{Diffeo}}^+ \, ({\mathbb R})$$ of order preserving diffeomorphisms. An element $$f$$ at such a group is called reversible (strongly reversible) if $$f^{-1} = h^{-1}fh$$ for any $$h$$ from the group (for any involution from the group). The main results are the following: (1) An element of $$\text{Diffeo} \, ({\mathbb R})$$ is reversible if and only if it is conjugate to a map $$f$$ from $$\text{Diffeo} \, ({\mathbb R})$$ that fixed each integer and satisfies the equation $$f(x + 1) = f^{-1}(x) + 1$$, $$x \in {\mathbb R}$$; (2) Each reversible element of $$\text{Diffeo} \, ({\mathbb R})$$ is either (i) strongly reversible, or (ii) an reversible (in $$\text{Diffeo}^+ \, ({\mathbb R})$$) element of $${\text{Diffeo}}^+ \, ({\mathbb R})$$. It is also considered the problem about representation of elements of the groups $$\text{Diffeo} \, ({\mathbb R})$$ as the composite of reversible diffeomorphisms or involutions. More precisely, it is proved that each element of $$\text{Diffeo} \, ({\mathbb R})$$ can be expressed as a compsite of four involutions and each element of $$\text{Diffeo}^+ \,({\mathbb R})$$ can be expressed as a composite of four reversible diffeomorphisms.

### MSC:

 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37E05 Dynamical systems involving maps of the interval 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems

### Keywords:

diffeomorphism; reversible; involution; conjugacy
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### References:

 [1] A. B. Calica, Reversible homeomorphisms of the real line, Pacific J. Math. 39 (1971), 79\Ndash87. · Zbl 0226.26024 [2] R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc. 218 (1976), 89\Ndash113. · Zbl 0363.58003 [3] N. J. Fine and G. E. Schweigert, On the group of homeomorphisms of an arc, Ann. of Math. (2) 62 (1955), 237\Ndash253. · Zbl 0066.41305 [4] W. Jarczyk, Reversible interval homeomorphisms, J. Math. Anal. Appl. 272(2) (2002), 473\Ndash479. · Zbl 1015.37031 [5] N. Kopell, Commuting diffeomorphisms, in: “Global Analysis” , Proc. Sympos. Pure Math. 14 , Amer. Math. Soc., Providence, R.I., 1970, pp. 165\Ndash184. · Zbl 0225.57020 [6] J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Time-reversal symmetry in dynamical systems (Coventry, 1996), Phys. D 112(1-2) (1998), 1\Ndash39. · Zbl 1194.34072 [7] J. Lubin, Non-Archimedean dynamical systems, Compositio Math. 94(3) (1994), 321\Ndash346. · Zbl 0843.58111 [8] A. G. O’Farrell, Conjugacy, involutions, and reversibility for real homeomorphisms, Irish Math. Soc. Bull. 54 (2004), 41\Ndash52. · Zbl 1076.57026 [9] A. G. O’Farrell, Composition of involutive power series, and reversible series, Comput. Methods Funct. Theory 8(1-2) (2008), 173\Ndash193. · Zbl 1232.20045 [10] A. G. O’Farrell and M. Roginskaya, Reducing conjugacy in the full diffeomorphism group of $$\mathbb{R}$$ to conjugacy in the subgroup of order preserving maps, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (Russian) 360 (2008), 231\Ndash237; translation in: J. Math. Sci. (N.Y.) 158 (2009), 895\Ndash898. · Zbl 1185.37035 [11] S. Sternberg, Local $$C^{n}$$ transformations of the real line, Duke Math. J. 24 (1957), 97\Ndash102. · Zbl 0077.06201 [12] S. W. Young, The representation of homeomorphisms on the interval as finite compositions of involutions, Proc. Amer. Math. Soc. 121(2) (1994), 605\Ndash610. · Zbl 0828.54013
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