Category:Mean Ergodic Theorem

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This category contains pages concerning Mean Ergodic Theorem:


Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $\map {L^2_\C} \mu$ be the complex-valued $L^2$ space of $\mu$.

Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be the Koopman operator.

Let $I := \set {f \in \map {L^2_\C} \mu : \map {U_T} f = f}$.


Then for each $f \in \map {L^2_\C} \mu$:

$\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} \map {U_T^n} f = \map {P_T} f$ converges in the $L^2$-norm

where:

$U_T^n$ denotes the $n$ times composition of $U_T$
$P_T: \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ denotes the orthogonal projection on the closed linear subspace $I$.

Pages in category "Mean Ergodic Theorem"

The following 5 pages are in this category, out of 5 total.