A direct method of constructing an invariant measure on a hyperbolic attractor

A new method of proving the existence of a natural invariant measure on a mixing hyperbolic attractor of a smooth mapping, and also its smooth dependence on the mapping, is proposed. It is proved directly that the sequence of mean integral values of a smooth function over the images of an arbitrary domain with a smooth measure converges with exponential speed to the mean value of the function with respect to an invariant measure. Here it is not required to construct a Markov partition, the expanding and contracting foliations, and the attractor itself.