Natural Invariant

Let rho(x)dx be the fraction of time a typical dynamical map orbit spends in the interval [x,x+dx], and let rho(x) be normalized such that


over the entire interval of the map. Then the fraction the time an map orbit spends in a finite interval [a,b], is given by


The natural invariant is also called the invariant density or natural density.

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Cite this as:

Weisstein, Eric W. "Natural Invariant." From MathWorld--A Wolfram Web Resource.

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