Geodesic
Certain Properties of Skew Products over a~Horseshoe and a~Solenoid
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 96-118

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The skew products are investigated over the Bernoulli shift and the Smale–Williams solenoid with a fiber $S^1$. It is assumed that the mapping in the fiber Hölder continuously depends on a point in the base (it is these skew products that arise in the study of partially hyperbolic sets). It is proved that, in the space of skew products with this property, there exists an open domain such that the mappings from this domain have dense sets of periodic orbits that are attracting and repelling along the fiber, as well as the dense orbits with the zero (along the fiber) Lyapunov exponent. 
@article{TM_2000_231_a3,
     author = {A. S. Gorodetski and Yu. S. Ilyashenko},
     title = {Certain {Properties} of {Skew} {Products} over {a~Horseshoe} and {a~Solenoid}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {96--118},
     publisher = {mathdoc},
     volume = {231},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2000_231_a3/}
}
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A. S. Gorodetski; Yu. S. Ilyashenko. Certain Properties of Skew Products over a~Horseshoe and a~Solenoid. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 96-118. http://geodesic.mathdoc.fr/item/TM_2000_231_a3/