TY - JOUR

T1 - Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

AU - Kruk, A. V.

AU - Malykh, A. E.

AU - Reitmann, V.

N1 - Funding Information: This work was supported by the Russian Science Foundation under grant no. 14-21-00041 by the German Academic Exchange Service (DAAD). Publisher Copyright: © 2017, Pleiades Publishing, Ltd. Copyright: Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85044100578&partnerID=8YFLogxK

U2 - 10.1134/S0012266117130031

DO - 10.1134/S0012266117130031

M3 - Article

AN - SCOPUS:85044100578

VL - 53

SP - 1715

EP - 1733

JO - Differential Equations

JF - Differential Equations

SN - 0012-2661

IS - 13

ER -