Halmos, "Lectures on ergodic theory" , Math. Japan MR Zbl Vershik, S.

## Ergodic Theorems of Birkhoff and von Neumann

Yuzvinskii, "Dynamical systems with an invariant measure" Progress in Math. Katok, Ya. Sinai, A. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" J. Soviet Math.

## ergodic hypothesis - oi

References [K] U. Encyclopedia of Mathematics. Von Neumann showed that a metrically transitive system will have properties whose average over time is equal to the average value of those properties over the phase space of all possible microstates.

It turns out that the weaker form [i. Von Neumann evidently planned to include his ergodic theorem and its proof in a much longer paper he was writing for the Annals of Mathematics , but he then apparently quickly drafted a short paper for PNAS with his proof of the mean ergodic theorem and submitted it to PNAS on December 10, It appeared in the January issue.

### 1. Dynamical Systems

In other words, the Quasi-Ergodic Hypothesis has been replaced by its modern version: the Hypothesis of Metrical Transitivity. These efforts to determine what types of systems might behave ergodically have come far since , but have a long way yet to go. See, e.

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Bellow and W. Proof of the Ergodic Theorem , Proc. Birkhoff and B. Recent Contributions to Ergodic Theory , Proc. Sci, U.

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Probability and Physical Systems , Bull Amer.