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Schoenflies Spheres as Boundaries of Bounded Unstable Manifolds in Gradient Sturm Systems

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Abstract

Let \(v\) be a hyperbolic equilibrium of a smooth finite-dimensional gradient or gradient-like dynamical system. Assume that the unstable manifold \(W\) of \(v\) is bounded, with topological boundary \(\Sigma = \partial \!W:= (clos W)\backslash W\). Then \(\Sigma \) need not be homeomorphic to a sphere, or to any compact manifold. However, consider PDEs

$$\begin{aligned} u_{t} = u_{xx} + f(x,u,u_x) \end{aligned}$$

of Sturm type, i.e. scalar reaction–advection–diffusion equations in one space dimension. Under separated boundary conditions on a bounded interval this defines a gradient dynamical system. For such gradient Sturm systems, we show that the eigenprojection P \(\Sigma \) of \(\Sigma \) onto the unstable eigenspace of \(v\) is homeomorphic to a sphere. In particular this excludes complications like lens spaces and Reidemeister torsion. Excluding Schoenflies complications like Alexander horned spheres, we also show that both the interior domain \(PW\) of P \(\Sigma \) and the one-point compactified exterior domain in the tangential eigenspace are homeomorphic to open balls. Our results are based on Sturm nodal properties.

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Acknowledgments

We are indebted to the late Floris Takens for cautioning us against the Reidemeister intricacy in the study of unstable manifolds of gradient systems. We are also grateful to Björn Sandstede and Matthias Wolfrum for sustained two-fold advice and encouragement: insisting that the problem was quite easy, but not spoiling our excitement by providing too many hints. The referee has helped with very diligent care and insight. Finally, we gratefully acknowledge mutual hospitality during extensive productive visits. This work was supported by the Deutsche Forschungsgemeinschaft, SFB 647 “Space–Time–Matter” and by FCT Portugal.

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  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 , Berlin, Germany

    Bernold Fiedler

  2. Departamento de Matemática, Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 , Lisbon, Portugal

    Carlos Rocha

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  1. Bernold Fiedler
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Correspondence to Bernold Fiedler.

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Dedicated to the memory of Klaus Kirchgässner.

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Fiedler, B., Rocha, C. Schoenflies Spheres as Boundaries of Bounded Unstable Manifolds in Gradient Sturm Systems. J Dyn Diff Equat 27, 597–626 (2015). https://doi.org/10.1007/s10884-013-9311-8

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