Geodesic
Foliated Functions and an Averaged Weighted Shift Operator for Perturbations of Hyperbolic Mappings
Informatics and Automation, 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). 
@article{TRSPY_2004_244_a3,
     author = {V. I. Bakhtin},
     title = {Foliated {Functions} and an {Averaged} {Weighted} {Shift} {Operator} for {Perturbations} of {Hyperbolic} {Mappings}},
     journal = {Informatics and Automation},
     pages = {35--64},
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     volume = {244},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2004_244_a3/}
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V. I. Bakhtin. Foliated Functions and an Averaged Weighted Shift Operator for Perturbations of Hyperbolic Mappings. Informatics and Automation, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 35-64. http://geodesic.mathdoc.fr/item/TRSPY_2004_244_a3/