ISSN:
1572-9222
Keywords:
Almost automorphy
;
ergodicity
;
scalar parabolic equations
;
minimal sets
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 ofH 2(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(Y 0)=1, whereμ is the unique ergodic measure of (Y, ℝ),Y 0={ity∈Y} Card(E∩P −1(y))=1},P:X × Y → Y is the natural projection (it was proved in an authors' earlier paper thatY 0 is a residual subset ofY). Moreover, if (E, ℝ) is uniquely ergodic, then it is topologically conjugated to a subflow ofR 1 ×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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02218894
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