Invariant measures of higher dimensional transformations are hard to calculate.
We present new results on estimation of absolutely continous invariant measures of mixing transformations, including a new method of proof of Ulam's conjecture.
The method involves constructing finite matrix approximations to the Perron-Frobenius operator from increasingly finer partitions of the state space.
We show that at a finite stage, our approximations are close to a special operator which would yield a correct answer.
The exponential mixing property guarantees that the system is sufficiently insensitive to any approximation errors, showing our computed invariant density is close to the true invariant density.
Our method has the advantages of having very relaxed conditions on the partitions, being applicable to higher dimensional systems, and potentially applicable to a wide class of maps.