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DEFINITION OF ERGODICITY Consider the "time average" of a Well-behaved function ''f''. This is defined as the average (if it exists) over iterations of ''T'' starting from some initial point ''x''. : Consider also the "space average" or "phase average" of ''f'', defined as : where μ is the measure of the probability space. In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time mean is equal to the space mean almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff . The Equidistribution Theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval. The time spent in a measurable set ''A'' is called the sojourn time. An immediate consequence of the ergodic theorem is that the measure of ''A'' is equal to the mean sojourn time. : where χ''A'' is the Indicator Function on ''A''. Let the occurrence times of a measurable set ''A'' be defined as the set ''k''1, ''k''2, ''k''3, ..., of times ''k'' such that ''T''''k''(''x'') is in ''A'', sorted in increasing order. The differences between consecutive occurrence times ''R''''i'' = ''k''''i'' − ''k''''i''−1 are called the recurrence times of ''A''. Another consequence of the ergodic theorem is that the average recurrence time of ''A'' is inversely proportional to the measure of ''A'', assuming that the initial point ''x'' is in ''A'', so that ''k''0 = 0. : (See Almost Surely .) That is, the smaller ''A'' is, the longer it takes to return to it. ERGODIC FLOWS ON MANIFOLDS The ergodicity of the Geodesic Flow on Manifolds Of Constant Negative Curvature was discovered by E. Hopf in 1939 . The relation between geodesic flows and one-parameter subgroups on SL(2,R) was given by S. V. Fomin and I. M. Gelfand in 1952 . Ergodicity of geodesic flow in Symmetric Space s was given by F. I. Mautner in 1957 . A simple criterion for the ergodicity of a homogeneous flow on a Homogeneous Space of a Semisimple Lie Group was given by C. C. Moore in 1966 . Many of the theorems and results from this area of study are typical of Rigidity Theory . The article on Anosov Flow s provides an example of ergodic flows on SL(2,R) and more generally on Riemann Surface s of negative curvature. Much of the development given there generalizes to Hyperbolic Manifold s of constant negative curvature, as these can be viewed as the quotient of a simply connected Hyperbolic Space modulo a Lattice in SO(n,1) . SEE ALSO Poincaré Recurrence Theorem HISTORICAL REFERENCES
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