Dyson showed that the Gaussian Unitary Ensemble (GUE) is the

invariant measure for this stochastic evolution, they explain, and conjectured that, when starting from a generic configuration of initial points, the time period that is needed for the GUE statistics to become dominant depends on the scale we look at: the microscopic correlations arrive at the equilibrium regime sooner than the macroscopic correlations.

Moreover, if P is an ergodic probability measure there is only one ergodic [theta]

invariant measure on ([[OMEGA].sub.P], F [intersection] [[OMEGA].sub.P] = {F [intersection] [[OMEGA].sub.P], F [member of] F}) (namely, P).

where we have used that [mu] is an

invariant measure so that [mu]([[PHI].sup.-1.sub.[tau]] = [mu](E).

The density function of an absolutely continuous

invariant measure with respect to the Lebesgue measure m of [0,1], often referred to as a stationary density, is a fixed point of the Frobenius-Perron operator PS : [L.sup.1](0,1) [right arrow] [L.sup.1](0,1) associated with the mapping.

The interspike-interval distribution [[mu].sub.ISI] with respect to the

invariant measure [mu] is defined as follows:

A Markov chain X with a transition kernel P and an

invariant measure [pi] is said to be v-strongly stable with respect to the norm [[parallel]*[parallel].sub.v] defined in 4), if [[parallel]P[parallel].sub.v] < [infinity] and each stochastic kernel Q on the space (N, B(N)) in some neighborhood {Q: [[parallel]Q - P[parallel].sub.v] < [epsilon]} has a unique

invariant measure [mu] = [mu](Q) and [[parallel][pi] - [mu][parallel].sub.v-] [right arrow] 0 as [[parallel] - P[parallel].sub.v] [right arrow] 0 in this neighborhood.

Then there exists an

invariant measure to (1) on C([S.sup.1]) for all [alpha] [greater than or equal to] 0, [c.sub.1] > 0, and [c.sub.2] > 0.

If the end user were to base its boundary on [P.sub.event] by using a rCS

invariant measure (a measure that could not reflect the end user's estimate of their attack rate), there would likely be either excessive false alarm processing costs or excessive expenses due to missed attacks.

Corollary 2 There exists a unique probability measure [bar.[mu]] [member of] [M.sub.1](X), called

invariant measure of the IFSP(w,p), such that M[bar.[mu]] = [bar.[mu]].

Looking at recent results in the area of ergodic theory (the mathematical study of dynamical systems with an

invariant measure) concerning the complexity of the problem of classification of ergodic measure preserving transformations up to conjugacy, the structure of the outer automorphism group of a countable measure preserving equivalence relation, ergodic theoretic characterizations with the Haagerup approximation property, and cocycle superrigidity, the author of this monograph realized that these apparently diverse results can all be understood within a unified framework.

(For [epsilon] = 0 this is usually called the Liouville equation.) Suppose that [[mu].sub.0] is an

invariant measure for (1.1) and the density [[rho].sub.0] of [[mu].sub.0] is a [C.sup.2] function.

functions for |[sigma]|ia set

invariant measures, relating different asymptotics of the "wave-packets" [v.sub.[alpha]] (N, [omega]) to the properties of the

invariant measure, and of its orthogonal polynomials.