Invariant measures for dissipative dynamical systems: abstract results and applications (Q694984)

Let \(S(t)\) be a continuous semigroup on a complete separable metric space \(X\), and let it be a global attractor of this semigroup. The authors show that for any generalized Banach limit and any probability distribution \(m_0\) of initial data there exists an invariant probability measure \(m\) with support in it such that \[ \int_X \varphi(x)\,dm(x)= \lim_{T\to\infty}{1\over T} \int^T_0 \int_X \varphi(S(t),x)\,dm_0(x)\,dt \] for all bounded continuous \(\varphi:X\to\mathbb{R}\). Hence, time averages and ensemble averages are equal.