A natural invariant, in the invariant-density convention for an interval map 
, is a probability
density function 
representing its long-term visit frequencies (Cvitanović et al. 2017,
pp. 353-354). More precisely, suppose 
is a measurable function
and 
is a nonnegative Lebesgue
integrable function on the interval 
. It defines a probability
measure
 |
(1)
|
for measurable sets 
, with normalization
 |
(2)
|
The measure must be an invariant measure, so 
for every such 
.
If 
is an ergodic measure, then for each fixed interval ![[a,b] subset= I](/images/equations/NaturalInvariant/Inline10.svg)
, Birkhoff's
ergodic theorem gives
![lim_(n->infty)(|{0<=j<n:T^j(x) in [a,b]}|)/n=int_a^brho(t)dt](/images/equations/NaturalInvariant/NumberedEquation3.svg) |
(3)
|
for 
-almost every 
. Thus the meaning of a typical map
orbit depends on the measure used. A physical
measure instead requires a set of initial points
of positive ambient Lebesgue measure whose averages
of continuous functions converge to that measure (Young 2002).
An invariant measure need not have a probability density function. For example, 
on ![[0,1]](/images/equations/NaturalInvariant/Inline14.svg)
has a physical measure 
with 
, since every map orbit
converges to zero. This probability measure
has no probability density function
with respect to Lebesgue measure, because the
singleton set 
has Lebesgue measure
zero. In the density convention, the natural invariant is also called the invariant
density or natural density.
