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Ergodicity of minimal sets in scalar parabolic equations

  • Wenxian Shen
  • Yingfei Yi
Article

Abstract

Skew product semiflowΠ t :X ×Y → X × Y generated by
$$\left\{ \begin{gathered} u_t = u_{xx} + f(y \cdot t,x,u,u_x ), t > 0 x \in (0,1), y \in Y, \hfill \\ D or N boundary conditions \hfill \\ \end{gathered} \right.$$
is considered, whereX is an appropriate subspace ofH2(0, 1), (Y, ℝ) is a compact minimal flow. By analyzing the zero crossing number for certain invariant manifolds and the linearized spectrum, it is shown that a minimal setE⊏X × Y ofΠ, is uniquely ergodic if and only if (Y, ℝ) is uniquely ergodic andμ(Y0)=1, whereμ is the unique ergodic measure of (Y, ℝ),Y0={ity∈Y} Card(E∩P−1(y))=1},P:X × Y → Y is the natural projection (it was proved in an authors' earlier paper thatY0 is a residual subset ofY). Moreover, if (E, ℝ) is uniquely ergodic, then it is topologically conjugated to a subflow ofR1 ×Y. A consequence of the last result is the following: in the case that (Y, ℝ) is almost periodic,Π, is expected to have many purely almost automorphic motions which are not ergodic.

Key words

Almost automorphy ergodicity scalar parabolic equations minimal sets 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Wenxian Shen
    • 1
  • Yingfei Yi
    • 2
  1. 1.Department of MathematicsAuburn UniversityAlabama
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaGeorgia

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