Geodesic
A direct method of constructing an invariant measure on a hyperbolic attractor
Izvestiya. Mathematics , Tome 41 (1993) no. 2, pp. 207-227

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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. 
@article{IM2_1993_41_2_a2,
     author = {V. I. Bakhtin},
     title = {A direct method of constructing an invariant measure on a hyperbolic attractor},
     journal = {Izvestiya. Mathematics },
     pages = {207--227},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1993_41_2_a2/}
}
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V. I. Bakhtin. A direct method of constructing an invariant measure on a hyperbolic attractor. Izvestiya. Mathematics , Tome 41 (1993) no. 2, pp. 207-227. http://geodesic.mathdoc.fr/item/IM2_1993_41_2_a2/