Differential Equations

, Volume 53, Issue 13, pp 1715–1733 | Cite as

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

Control Theory


We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.


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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsSaint Petersburg State UniversityPeterhofRussia

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