Geodesic
Shadowing in linear skew products
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 261-273

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We consider a linear skew product with the full shift in the base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of shadowing for a typical pseudotrajectory of finite length. This result indicates that the high-dimensional analog of the Hammel–Yorke–Grebogi conjecture concerning the interval of shadowability for a typical pseudotrajectory is not correct. The main technique is the reduction of the shadowing problem to the ruin problem for a simple random walk. 
@article{ZNSL_2015_432_a13,
     author = {S. Tikhomirov},
     title = {Shadowing in linear skew products},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {261--273},
     publisher = {mathdoc},
     volume = {432},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a13/}
}
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S. Tikhomirov. Shadowing in linear skew products. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 261-273. http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a13/