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Invariant measures for complexvalued dissipative dynamical systems and applications
1.  School of Mathematical and Statistics, Lanzhou University, Lanzhou, Gansu, China 
2.  Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA 
In this work, we extend the classical realvalued framework to deal with complexvalued dissipative dynamical systems. With our new complexvalued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data $u_{0}$, we construct a unique complex invariant measure $\mu$ on a metric space which is acted by a continuous semigroup $\{S(t)\}_{t\geq 0}$ possessing a global attractor $\mathcal{A}$. Moreover, it is shown that the support of $\mu$ is not only contained in global attractor $\mathcal{A}$ but also in $\omega(u_{0})$. Next, the structure of the measure $\mu$ is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex GinzburgLandau equation and a nonlinear Schrödinger equation and construct complex invariant measures for these two complexvalued equations.
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