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The New Palgrave Dictionary of Economics pp 3878–3884Cite as

Palgrave Macmillan

Ergodic Theory

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Abstract

To begin in the middle; for that is where ergodic theory started, in the middle of the development of statistical mechanics, with the solution, by von Neumann and Birkhoff, of the problem of identifying space averages with time averages. This problem can be formulated as follows: If xI(− ∞ < t < ∞) represents the trajectory (orbit) passing through the point x = x0 at time t = 0 of a conservative dynamical system, when can one make the identification

$$ \left({}^{\ast}\right) \lim \limits_{T\to \infty } \left(1/T\right){\int}_0^Tf\left({x}_t\right)\mathrm{d}t={\int}_{\Omega}f\;\mathrm{d}m/m\left(\Omega \right) $$

for suitable functions defined on the phase space Ω of the system?

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  1. http://link.springer.com/referencework/10.1007/978-1-349-95121-5

    William Parry

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Parry, W. (2018). Ergodic Theory. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_208

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