Osaka Journal of Mathematics

Local inverses of shift maps along orbits of flows

Sergiy Maksymenko

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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

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

First available in Project Euclid: 6 September 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C05: Smooth mappings and diffeomorphisms 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms 57R45: Singularities of differentiable mappings


Maksymenko, Sergiy. Local inverses of shift maps along orbits of flows. Osaka J. Math. 48 (2011), no. 2, 415--455.

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