Osaka Journal of Mathematics

Local inverses of shift maps along orbits of flows

Sergiy Maksymenko

Abstract

Let $\mathbf{F}$ be a smooth flow on a smooth manifold $M$ and $\mathcal{D}(\mathbf{F})$ be the group of diffeomorphisms of $M$ preserving orbits of $\mathbf{F}$. We study the homotopy type of the identity components $\mathcal{D}_{\id}(\mathbf{F})^{r}$ of $\mathcal{D}(\mathbf{F})$ with respect to distinct Whitney topologies $\mathsf{W}^{r}$, ($0 \leq r \leq \infty$). The main result presents a class of flows $\mathbf{F}$ for which $\mathcal{D}_{\id}(\mathbf{F})^{r}$ coincide for all $r$ and are either contractible or homotopy equivalent to the circle. The group $\mathcal{D}_{\id}(\mathbf{F})^{0}$ was studied in the author's paper [13]. Unfortunately that article contains a gap in estimations of continuity of local inverses of the so-called shift map. The present paper also repairs these estimations and shows that they hold under additional assumptions on the behavior of regular points of $\mathbf{F}$.

Article information

Source
Osaka J. Math., Volume 48, Number 2 (2011), 415-455.

Dates
First available in Project Euclid: 6 September 2011

https://projecteuclid.org/euclid.ojm/1315318347

Mathematical Reviews number (MathSciNet)
MR2831980

Zentralblatt MATH identifier
1291.37026

Citation

Maksymenko, Sergiy. Local inverses of shift maps along orbits of flows. Osaka J. Math. 48 (2011), no. 2, 415--455. https://projecteuclid.org/euclid.ojm/1315318347

References

• D.L. Blackmore: On the local normalization of a vector field at a degenerate critical point, J. Differential Equations 14 (1973), 338–359.
• A.D. Brjuno: Analytic form of differential equations, I, Trudy Moskov. Mat. Obšč. 25 (1971), 119–262.
• R.V. Chacon: Change of velocity in flows, J. Math. Mech. 16 (1966), 417–431.
• C. Conley and R. Easton: Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61.
• L. Frerick: Extension operators for spaces of infinite differentiable Whitney jets, J. Reine Angew. Math. 602 (2007), 123–154.
• D. Hart: On the smoothness of generators, Topology 22 (1983), 357–363.
• M.W. Hirsch: Differential Topology, corrected reprint of the 1976 original, Graduate Texts in Mathematics 33, Springer, New York, 1994.
• E. Hopf: Ergodentheorie, Chelsea Publishing Co., Berlin, 1937.
• S. Illman: The very-strong $C^{\infty}$ topology on $C^{\infty}(M,N)$ and $K$-equivariant maps, Osaka J. Math. 40 (2003), 409–428.
• A.V. Kočergin: Change of time in flows, and mixing, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1275–1298.
• V.A. Kondrat\textsuperscript$\prime$ev and V.S. Samovol: The linearization of an autonomous system in the neighborhood of a singular point of “knot” type, Mat. Zametki 14 (1973), 833–842.
• M. Kowada: The orbit-preserving transformation groups associated with a measurable flow, J. Math. Soc. Japan 24 (1972), 355–373.
• S. Maksymenko: Smooth shifts along trajectories of flows, Topology Appl. 130 (2003), 183–204.
• S. Maksymenko: Hamiltonian vector fields of homogeneous polynomials in two variables, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos. 3 (2006), 269–308, arXiv:math/0709.2511.
• S. Maksymenko: Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom. 29 (2006), 241–285.
• S. Maksymenko: Stabilizers and orbits of smooth functions, Bull. Sci. Math. 130 (2006), 279–311.
• S. Maksymenko: Connected components of partition preserving diffeomorphisms, Methods Funct. Anal. Topology 15 (2009), 264–279.
• S. Maksymenko: Image of a shift map along the orbits of a flow, to appear in Indiana Univ. Math. J.
• S. Maksymenko: $\infty$-jets of diffeomorphisms preserving orbits of vector fields, Cent. Eur. J. Math. 7 (2009), 272–298.
• S. Maksymenko: Reparametrization of vector fields and their shift maps, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos. 6 (2009), 489–498, arXiv:math/0907.0354.
• S. Maksymenko: Symmetries of center singularities of plane vector fields, Nonlinear Oscilations 13 (2010), 177–205.
• J. Margalef Roig and E. Outerelo Domí nguez: Differential Topology, North-Holland Mathematics Studies 173, North-Holland, Amsterdam, 1992.
• J.N. Mather: Stability of $C^{\infty}$ mappings, I, The division theorem, Ann. of Math. (2) 87 (1968), 89–104.
• B.S. Mitjagin: Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961), 63–132.
• M.A. Mostow and S. Shnider: Joint continuity of division of smooth functions, I, Uniform Lojasiewicz estimates, Trans. Amer. Math. Soc. 292 (1985), 573–583.
• J. Palis, Jr. and W. de Melo: Geometric Theory of Dynamical Systems, translated from the Portuguese by A.K. Manning, Springer, New York, 1982.
• W. Parry: Cocycles and velocity changes, J. London Math. Soc. (2) 5 (1972), 511–516.
• R.T. Seeley: Extension of $C^{\infty}$ functions defined in a half space, Proc. Amer. Math. Soc. 15 (1964), 625–626.
• C.L. Siegel: Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1952 (1952), 21–30.
• S. Sternberg: Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824.
• H. Totoki: Time changes of flows, Mem. Fac. Sci. Kyushu Univ. Ser. A 20 (1966), 27–55.
• R.J. Venti: Linear normal forms of differential equations, J. Differential Equations 2 (1966), 182–194.
• H. Whitney: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89.