Natural Invariant

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

$$\int_0^\infty \rho(x)\,dx=1$$

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

$$\int_a^b \rho(x)\,dx.$$

The natural invariant is also called the Invariant Density or Natural Density.