Geodesic
Foliated Functions and an Averaged Weighted Shift Operator for Perturbations of Hyperbolic Mappings
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 35-64

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In order to study the perturbations of a family of mappings with a hyperbolic mixing attractor, an apparatus of foliated functions is developed. Foliated functions are analogues of distributions based on smooth measures on leaves (traces), which are embedded manifolds in a neighborhood of the attractor. The dimension of such manifolds must coincide with the dimension of the expanding foliation, and the values of a foliated function on a trace must vary smoothly under smooth transverse deformations of the trace (which include deformations of the measure itself). 
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     author = {V. I. Bakhtin},
     title = {Foliated {Functions} and an {Averaged} {Weighted} {Shift} {Operator} for {Perturbations} of {Hyperbolic} {Mappings}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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V. I. Bakhtin. Foliated Functions and an Averaged Weighted Shift Operator for Perturbations of Hyperbolic Mappings. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 35-64. http://geodesic.mathdoc.fr/item/TM_2004_244_a3/